Document:

<PAGE>

                                                                    EXHIBIT 4.04

                          CERTIFICATE OF DESIGNATION

                                      of

                SERIES B NON-VOTING CONVERTIBLE PREFERRED STOCK

                                      of

                              AT HOME CORPORATION

                        (Pursuant to Section 151 of the
                       Delaware General Corporation Law)

                       ---------------------------------

     At Home Corporation, a Delaware corporation (the "Company"), does hereby
certify that pursuant to the authority granted to and vested in the Board of
Directors of the Company (the "Board of Directors" or the "Board") in accordance
with the provisions of Article IV, Section C of the Fifth Amended and Restated
Certificate of Incorporation of the Company, and in accordance with the
provisions of Section 151 of the Delaware General Corporation Law, the Company's
Board of Directors has duly adopted the following resolutions creating a series
of Preferred Stock, par value $0.01 per share ("Preferred Stock"), of the
Company designated as Series B Non-Voting Convertible Preferred Stock.

     RESOLVED, that the Company hereby designates and creates a series of
Preferred Stock of the Company and hereby states the designation and number of
shares, and fixes the relative rights, preferences, and limitations thereof as
follows:

     A.  Designation and Amount.  Of the 9,650,000 shares of Preferred Stock
         ----------------------
authorized to be issued by the Company, 1,008 shares shall be designated as
"Series B Non-Voting Convertible Preferred Stock" ("Series B Preferred Stock").
Such number of shares may be increased or decreased by resolution of the Board
of Directors; provided, that no decrease shall reduce the number of shares of
              --------
Series B Preferred Stock to a number less than the number of shares then
outstanding plus the number of shares reserved for issuance upon the exercise of
outstanding options, rights or warrants or upon the conversion of any
outstanding securities issued by the Company convertible into Series B Preferred
Stock.

     B.  Rights, Preferences and Restrictions of Preferred Stock.  The rights,
         -------------------------------------------------------
preferences, restrictions and other matters relating to the Series B Preferred
Stock are as follows:

         1.  Dividends and Distributions.
             ---------------------------

             (a)  Subject to the rights of the holders of any shares of any
series of Preferred Stock (or any other stock) ranking prior and superior to the
Series B Preferred Stock with respect to dividends, and subject to Section
5(c)(ii) hereof, the holders of shares of Series B Preferred Stock shall be
entitled to receive, when, as and if declared by the Board of Directors out of
funds legally available for the purpose, non-cumulative dividends in an amount
(if any) per share (rounded to the nearest cent), subject to the provision for
adjustment hereinafter set forth, equal to 1,000 times the aggregate per share
amount of all cash dividends, and 1,000 times the aggregate per share amount
(payable in kind) of all non-cash dividends or other distributions, other than a
dividend payable in shares of Series A Common Stock, par value $0.01 per share
(the "Common Stock") of the Company or other distribution payable in securities,
or a subdivision of the outstanding shares of Common Stock (by reclassification
or otherwise), declared on the Common Stock. In the event the Company shall at
any time declare or pay any dividend on the
<PAGE>

Common Stock payable in shares of Common Stock, or effect a subdivision or
combination or consolidation of the outstanding shares of Common Stock (by
reclassification or otherwise than by payment of a dividend in shares of Common
Stock) into a greater or lesser number of shares of Common Stock, then in each
such case the amount to which such holders of shares of Series B Preferred Stock
were entitled to immediately prior to such event under the preceding sentence
shall be adjusted by multiplying such amount by a fraction, the numerator of
which is the number of shares of Common Stock outstanding immediately after such
event and the denominator of which is the number of shares of Common Stock that
were outstanding immediately prior to such event.

             (b)  The Company shall declare a dividend or distribution on the
Series B Preferred Stock as provided in subsection (a) of this Section at the
same time it declares a dividend or distribution on the Common Stock (other than
a dividend or distribution payable in shares of Common Stock or other
distribution payable in securities). The Board of Directors shall fix a record
date for the determination of holders of shares of Series B Preferred Stock
entitled to receive payment of such dividend or distribution declared thereon,
which record date shall be not more than 60 days prior to the date fixed for the
payment thereof and shall be on the same day as the record date fixed for the
Common Stock.

             (c)  Dividends payable pursuant to subsection (a) of this Section
shall not be mandatory or cumulative (except as provided in subsection (b) of
this Section) and no rights or interest shall accrue to the holders of the
Series B Preferred Stock by reason of the fact that the Company shall fail to
declare or pay dividends on the Preferred Stock in any amount in any calendar
year or any fiscal year of the Company, whether or not earnings of the Company
in any calendar year or fiscal year were sufficient to pay such dividends in
whole or in part.

             (d)  Except as provided in this Section 1, holders of the Series B
Preferred Stock shall not be entitled to receive any dividends.

         2.  Voting Rights.  Except as otherwise provided by law, holders of
             -------------
Series B Preferred Stock shall have no voting rights and their consent shall not
be required for taking any corporate action.

         3.  Reacquired and Converted Shares. Any shares of Series B Preferred
             -------------------------------
Stock purchased, received upon conversion or otherwise acquired by the Company
in any manner whatsoever shall be retired and canceled promptly after the
acquisition thereof and shall not be reissued.

         4.  Liquidation, Dissolution or Winding Up.
             --------------------------------------

             (a)  Upon any liquidation, dissolution or winding up of the
Company, the holders of shares of Series B Preferred Stock shall be entitled to
receive an aggregate amount per share, subject to the provision for adjustment
hereinafter set forth, equal to 1,000 times the aggregate amount to be
distributed per share to holders of shares of Common Stock, on a pro rata pari
passu basis with the Common Stock, plus an amount equal to any declared but
unpaid dividends. In the event the Company shall at any time declare or pay any
dividend on the Common Stock payable in shares of Common Stock, or effect a
subdivision or combination or consolidation of the outstanding shares of Common
Stock (by reclassification or otherwise than by payment of a dividend in shares
of Common Stock) into a greater or lesser number of shares of Common Stock, then
in each such case the aggregate amount to which such holders of shares of Series
B Preferred Stock were entitled to immediately prior to such event under the
preceding sentence shall be adjusted by multiplying such amount by a fraction
the numerator of which is the number of shares of Common Stock outstanding
immediately after
<PAGE>

such event and the denominator of which is the number of shares of Common Stock
that were outstanding immediately prior to such event.

             (b)  The foregoing rights upon liquidation, dissolution or winding
up provided to the holders of Series B Preferred Stock shall be subject to the
rights of the holders of any other series of Preferred Stock (or any other
stock) ranking prior and superior to the Series B Preferred Stock upon
liquidation, dissolution or winding up.

         5.  Conversion.
             ----------

             (a)  Automatic Conversion. On the Conversion Date, each outstanding
                  --------------------
shares of Series B Preferred Stock shall automatically convert into the right to
receive that number of shares of Common Stock as equals the Common Stock
Conversion Number.

             (b)  Mechanics of Conversion. Upon the conversion of the Series B
                  -----------------------
Preferred Stock as specified in Section 5(a) above, the outstanding shares of
Series B Preferred Stock shall be converted into shares of Common Stock
automatically without the need for any further action by the holders of such
shares and whether or not the certificates representing such shares are
surrendered to the Company or its transfer agent; provided, however,
                                                  --------  -------
that the Company shall not be obligated to issue certificates evidencing such
shares of Common Stock issuable upon such conversion unless the certificates
evidencing such shares of Series B Preferred Stock are either delivered to the
Company or its transfer agent as provided below, or the holder notifies the
Company or its transfer agent that such certificates have been lost, stolen or
destroyed and executes an agreement satisfactory to the Company to indemnify the
Company from any loss incurred by it in connection with such certificates.
Following the occurrence of such automatic conversion of the Series B Preferred
Stock, the holders of Series B Preferred Stock shall surrender the certificates
representing such shares at the office of the Company or any transfer agent for
the Series B Preferred Stock. Thereupon, there shall be issued and delivered to
such holder promptly at such office and in its name as shown on such surrendered
certificate or certificates, a certificate or certificates for the number of
shares of Common Stock into which the shares of Series B Preferred Stock
surrendered were converted on the date on which such automatic conversion
occurred. Such conversion shall be deemed to have been made immediately prior to
the close of business on the Conversion Date, and the person or persons entitled
to receive the shares of Common Stock issuable upon such conversion shall be
treated for all purposes as the record holder or holders of such shares of
Common Stock on such date.

             (c)  Adjustments to Common Stock Conversion Number for Certain
                  ---------------------------------------------------------
Diluting Issues.
---------------

                  (i)     Adjustments for Dividends, Splits, Subdivisions,
                          ------------------------------------------------
Combinations or Consolidation of Common Stock. In the event the outstanding
---------------------------------------------
shares of Common Stock shall be increased by stock dividend payable in Common
Stock, stock split, subdivision, or other similar transaction into a greater
number of shares of Common Stock, the Common Stock Conversion Number of the
Series B Preferred Stock then in effect shall, concurrently with the
effectiveness of such event, be increased proportionally. In the event the
outstanding shares of Common Stock shall be decreased by reverse stock split,
combination, consolidation, or other similar transaction into a lesser number of
shares of Common Stock, the Common Stock Conversion Number of the Series B
Preferred Stock then in effect shall, concurrently with the effectiveness of
such event, be decreased proportionally.

                  (ii)    Adjustments for Other Distributions. In the event the
                          -----------------------------------
Company at any time or from time to time makes, or fixes a record date for the
determination of holders of Common Stock entitled to receive any distribution
payable in securities (other than shares of Common Stock and other than any
distribution which results in the payment of a dividend
<PAGE>

described in Section 1), then, and in each such event, provision shall be made
so that the holders of Series B Preferred Stock shall receive upon conversion
thereof, in addition to the number of shares of Common Stock receivable
thereupon, the amount of securities of the Company which they would have
received had their Series B Preferred Stock been converted into Common Stock
immediately prior to the date of such event and had they thereafter, during the
period immediately prior to the date of such event to and including the date of
conversion, retained such securities receivable by them as aforesaid during such
period, subject to all other adjustments called for during such period under
this Section 5 with respect to the rights of the holders of the Series B
Preferred Stock.

                  (iii)   Adjustments for Reclassification, Exchange and
                          ----------------------------------------------
Substitution. If the Common Stock issuable upon conversion of the Series B
------------
Preferred Stock shall be changed into the same or a different number of shares
of any other class or classes of stock, whether by capital reorganization,
reclassification, or otherwise (other than a subdivision or combination of
shares provided for above), then following the effectiveness of such
reorganization or reclassification the Series B Preferred Stock shall be
convertible into, in lieu of the number of shares of Common Stock which the
holders would otherwise have been entitled to receive, a number of shares of
such other class or classes of stock equivalent to the number of shares of such
other class or classes of stock into which the Common Stock was converted in
such reorganization or reclassification that would have been subject to receipt
by the holders upon conversion of such Series B Preferred Stock immediately
before that change.

             (d)  No Impairment. Except as provided in Section 5, the Company
                  -------------
will not, by amendment of its Certificate of Incorporation or through any
reorganization, transfer of assets, consolidation, merger, dissolution, issue or
sale of securities, or any other voluntary action, avoid or seek to avoid the
observance or performance of any of the terms to be observed or performed
hereunder by the Company but will at all times in good faith assist in the
carrying out of all the provisions of this Section 5 and in the taking of all
such action as may be necessary or appropriate in order to protect the
conversion rights of the holders of the Series B Preferred Stock against
impairment.

             (e)  Certificate as to Adjustments of the Common Stock Conversion
                  ------------------------------------------------------------
Number. Upon the occurrence of each adjustment or readjustment of the Common
-----
Stock Conversion Number for the Series B Preferred Stock pursuant to this
Section 5, the Company at its expense shall promptly compute such adjustment or
readjustment in accordance with the terms hereof and furnish to each holder of
Series B Preferred Stock a certificate setting forth such adjustment or
readjustment and showing in detail the facts upon which such adjustment or
readjustment is based. The Company shall, upon the written request at any time
of any holder of Series B Preferred Stock, furnish or cause to be furnished to
such holder a like certificate setting forth (i) such adjustments and
readjustments, (ii) the Common Stock Conversion Number at the time in effect,
and (iii) the number of shares of Common Stock and the amount, if any, of other
property which at the time would be received upon the conversion of the Series B
Preferred Stock.

             (f)  Notices of Record Date. In the event that this Company shall
                  ----------------------
propose at any time:

                  (i)     to declare any dividend or distribution upon its
Common Stock, whether in cash, property, stock, or other securities, whether or
not a regular cash dividend and whether or not out of earnings or earned
surplus;

                  (ii)    to offer for subscription pro rata to the holders of
any class or series of its stock any additional shares of stock of any class or
series or other rights;
<PAGE>

                  (iii)   to effect any reclassification or recapitalization of
its Common Stock outstanding involving a change in the Common Stock; or

                  (iv)    to merge or consolidate with or into any other
corporation, or sell, lease, or convey all or substantially all its property or
business, or to liquidate, dissolve, or wind up; then, in connection with each
such event, this Company shall send to the holders of Series B Preferred Stock:

                          (1)  at least 10 days prior written notice of the date
on which a record shall be taken for such dividend, distribution or subscription
rights (and specifying the date on which the holders of Common Stock shall be
entitled thereto) or for determining rights to vote in respect of the matters
referred to in (iii) and (iv) above; and

                          (2)  in the case of the matters referred to in (iii)
and (iv) above, at least 20 days' prior written notice of the date when the same
shall take place (and specifying the date on which the holders of Common Stock
shall be entitled to exchange their Common Stock for securities or other
property deliverable upon the occurrence of such event or the record date for
the determination of such holders if such record date is earlier).

     Each such written notice shall be delivered personally, or given by first
class mail, postage prepaid, addressed to the holders of the Series B Preferred
Stock at the address for each such holder as shown on the books of this Company.

             (g)  Issue Taxes. The Company shall pay any and all issue and other
                  -----------
taxes (other than income taxes) that may be payable in respect of any issue or
delivery of shares of Common Stock on conversion of shares of Series B Preferred
Stock pursuant hereto; provided, however, that the Company shall not be
obligated to pay any transfer taxes resulting from any transfer requested by any
holder in connection with any such conversion.

             (h)  Reservation of Stock Issuable Upon Conversion. At the
                  ---------------------------------------------
Conversion Date, the Company shall have reserved and have available out of its
authorized but unissued shares of Common Stock, solely for the purpose of
effecting the conversion of the shares of the Series B Preferred Stock such
number of its shares of Common Stock as shall be sufficient to effect the
conversion of all outstanding shares of Series B Preferred Stock; and if at any
time the number of authorized but unissued shares of Common Stock shall not be
sufficient to effect the conversion of all then outstanding shares of the Series
B Preferred Stock, the Company will take such corporate action as may, in the
opinion of its counsel, be necessary to increase its authorized but unissued
shares of Common Stock to such number of shares as shall be sufficient for such
purpose, including, without limitation, engaging in best efforts to obtain the
requisite stockholder approval of any necessary amendment to its Certificate of
Incorporation.

             (i)  Defined Terms
                  -------------

     As used herein, the following terms as have the meanings set forth below:

             "Conversion Date," as applied to any shares of Series B Preferred
Stock, shall mean the date which is the one year anniversary of the date on
which such shares of Series B Preferred Stock were originally issued.

             "Common Stock Conversion Number" shall mean one thousand (1000)
shares of Common Stock, as adjusted pursuant to Section 5(c) hereof.

         6.  Consolidation, Merger, etc. In case the Company shall enter into
             --------------------------
any consolidation, merger, combination or other transaction in which the shares
of Common Stock are exchanged for or changed into other stock or securities,
cash and/or other property, then in any such case each share of Series B
Preferred Stock shall at the same time be similarly exchanged or changed into an
amount per share, subject to the provision for adjustment hereinafter set forth,
<PAGE>

equal to 1,000 times the aggregate amount of stock, securities, cash and/or any
other property (payable in kind), as the case may be, into which or for which
each share of Common Stock is changed or exchanged. In the event the Company
shall at any time declare or pay any dividend on the Common Stock payable in
shares of Common Stock, or effect a subdivision or combination or consolidation
of the outstanding shares of Common Stock (by reclassification or otherwise than
by payment of a dividend in shares of Common Stock) into a greater or lesser
number of shares of Common Stock, then in each such case the amount set forth in
the preceding sentence with respect to the exchange or change of shares of
Series B Preferred Stock shall be adjusted by multiplying such amount by a
fraction, the numerator of which is the number of shares of Common Stock
outstanding immediately after such event and the denominator of which is the
number of shares of Common Stock that were outstanding immediately prior to such
event.

         7.  No Redemption. The shares of Series B Preferred Stock shall not be
             -------------
redeemable.

     IN WITNESS WHEREOF, this Certificate of Designation is executed on behalf
of the Company this 9th day of February, 2000.

                                            AT HOME CORPORATION

                                            By: /s/ David Pine
                                               ---------------------------
                                               David Pine, Secretary<PAGE>

                                                                    EXHIBIT 4.05

                          CERTIFICATE OF DESIGNATION
                                      of
                SERIES C NON-VOTING CONVERTIBLE PREFERRED STOCK
                                      of
                              AT HOME CORPORATION
                        (Pursuant to Section 151 of the
                       Delaware General Corporation Law)

                       ---------------------------------
     At Home Corporation, a Delaware corporation (the "Company"), does hereby
certify that pursuant to the authority granted to and vested in the Board of
Directors of the Company (the "Board of Directors" or the "Board") in accordance
with the provisions of Article IV, Section C of the Fifth Amended and Restated
Certificate of Incorporation of the Company, and in accordance with the
provisions of Section 151 of the Delaware General Corporation Law, the Company's
Board of Directors has duly adopted the following resolutions creating a series
of Preferred Stock, par value $0.01 per share ("Preferred Stock"), of the
Company designated as Series C Non-Voting Convertible Preferred Stock.

     RESOLVED, that the Company hereby designates and creates a series of
Preferred Stock of the Company and hereby states the designation and number of
shares, and fixes the relative rights, preferences, and limitations thereof as
follows:

     A.  Designation and Amount.  Of the 9,650,000 shares of Preferred Stock
         ----------------------
authorized to be issued by the Company, 1,279 shares shall be designated as
"Series C Non-Voting Convertible Preferred Stock" ("Series C Preferred Stock").
Such number of shares may be increased or decreased by resolution of the Board
of Directors; provided, that no decrease shall reduce the number of shares of
              --------
Series C Preferred Stock to a number less than the number of shares then
outstanding plus the number of shares reserved for issuance upon the exercise of
outstanding options, rights or warrants or upon the conversion of any
outstanding securities issued by the Company convertible into Series C Preferred
Stock.

     B.  Rights, Preferences and Restrictions of Preferred Stock.  The rights,
         -------------------------------------------------------
preferences, restrictions and other matters relating to the Series C Preferred
Stock are as follows:

         1.  Dividends and Distributions.
             ---------------------------
             (a)    Subject to the rights of the holders of any shares of any
series of Preferred Stock (or any other stock) ranking prior and superior to the
Series C Preferred Stock with respect to dividends, and subject to Section
5(c)(ii) hereof, the holders of shares of Series C Preferred Stock shall be
entitled to receive, when, as and if declared by the Board of Directors out of
funds legally available for the purpose, non-cumulative dividends in an amount
(if any) per share (rounded to the nearest cent), subject to the provision for
adjustment hereinafter set forth, equal to 1,000 times the aggregate per share
amount of all cash dividends, and 1,000 times the aggregate per share amount
(payable in kind) of all non-cash dividends or other distributions, other than a
dividend payable in shares of Series A Common Stock, par value $0.01 per share
(the "Common Stock") of the Company or other distribution payable in securities,
or a subdivision of the outstanding shares of Common Stock (by reclassification
or otherwise), declared on the Common Stock. In the event the Company shall at
any time declare or pay any dividend on the Common Stock payable in shares of
Common Stock, or effect a subdivision or combination or consolidation of the
outstanding shares of Common Stock (by reclassification or otherwise than by
payment of a dividend in shares of Common Stock) into a greater or lesser number
of shares of Common Stock, then in each such case the amount to which such
holders of shares of Series C Preferred Stock were entitled to immediately prior
to such event under the preceding sentence shall be adjusted by multiplying such
amount by a fraction, the numerator of which is the number of shares of Common
Stock outstanding immediately after such event and the denominator of which is
the number of shares of Common Stock that were outstanding immediately prior to
such event.
<PAGE>

             (b)    The Company shall declare a dividend or distribution on the
Series C Preferred Stock as provided in subsection (a) of this Section at the
same time it declares a dividend or distribution on the Common Stock (other than
a dividend or distribution payable in shares of Common Stock or other
distribution payable in securities). The Board of Directors shall fix a record
date for the determination of holders of shares of Series C Preferred Stock
entitled to receive payment of such dividend or distribution declared thereon,
which record date shall be not more than 60 days prior to the date fixed for the
payment thereof and shall be on the same day as the record date fixed for the
Common Stock.

             (c)    Dividends payable pursuant to subsection (a) of this Section
shall not be mandatory or cumulative (except as provided in subsection (b) of
this Section) and no rights or interest shall accrue to the holders of the
Series C Preferred Stock by reason of the fact that the Company shall fail to
declare or pay dividends on the Preferred Stock in any amount in any calendar
year or any fiscal year of the Company, whether or not earnings of the Company
in any calendar year or fiscal year were sufficient to pay such dividends in
whole or in part.

             (d)    Except as provided in this Section 1, holders of the Series
C Preferred Stock shall not be entitled to receive any dividends.

         2.  Voting Rights.  Except as otherwise provided by law, holders of
             -------------
Series C Preferred Stock shall have no voting rights and their consent shall not
be required for taking any corporate action.

         3.  Reacquired and Converted Shares. Any shares of Series C Preferred
             -------------------------------
Stock purchased, received upon conversion or otherwise acquired by the Company
in any manner whatsoever shall be retired and canceled promptly after the
acquisition thereof and shall not be reissued.

         4.  Liquidation, Dissolution or Winding Up.
             --------------------------------------
             (a)    Upon any liquidation, dissolution or winding up of the
Company, the holders of shares of Series C Preferred Stock shall be entitled to
receive an aggregate amount per share, subject to the provision for adjustment
hereinafter set forth, equal to 1,000 times the aggregate amount to be
distributed per share to holders of shares of Common Stock, on a pro rata pari
passu basis with the Common Stock, plus an amount equal to any declared but
unpaid dividends. In the event the Company shall at any time declare or pay any
dividend on the Common Stock payable in shares of Common Stock, or effect a
subdivision or combination or consolidation of the outstanding shares of Common
Stock (by reclassification or otherwise than by payment of a dividend in shares
of Common Stock) into a greater or lesser number of shares of Common Stock, then
in each such case the aggregate amount to which such holders of shares of Series
C Preferred Stock were entitled to immediately prior to such event under the
preceding sentence shall be adjusted by multiplying such amount by a fraction
the numerator of which is the number of shares of Common Stock outstanding
immediately after such event and the denominator of which is the number of
shares of Common Stock that were outstanding immediately prior to such event.

             (b)    The foregoing rights upon liquidation, dissolution or
winding up provided to the holders of Series C Preferred Stock shall be subject
to the rights of the holders of any other series of Preferred Stock (or any
other stock) ranking prior and superior to the Series C Preferred Stock upon
liquidation, dissolution or winding up.

         5.  Conversion.
             ----------
             (a)    Automatic Conversion.  On the Conversion Date, each
                    --------------------
outstanding share of Series C Preferred Stock shall automatically convert into
the right to receive that number of shares of Common Stock as equals the Common
Stock Conversion Number.
             (b)    Mechanics of Conversion. Upon the conversion of the Series C
                    -----------------------
Preferred Stock as specified in Section 5(a) above, the outstanding shares of
Series C Preferred Stock shall be converted into shares of Common Stock
automatically without the need for any further action by the holders of such
shares and whether or not the certificates representing such shares are
surrendered to the Company or its transfer agent; provided, however,
                                                  --------  -------
that the Company shall not be obligated to issue certificates evidencing such
shares of Common Stock issuable upon such conversion unless the certificates
evidencing such shares of Series C Preferred Stock are either delivered to the
Company or its

                                      D-2
<PAGE>

transfer agent as provided below, or the holder notifies the Company or its
transfer agent that such certificates have been lost, stolen or destroyed and
executes an agreement satisfactory to the Company to indemnify the Company from
any loss incurred by it in connection with such certificates. Following the
occurrence of such automatic conversion of the Series C Preferred Stock, the
holders of Series C Preferred Stock shall surrender the certificates
representing such shares at the office of the Company or any transfer agent for
the Series C Preferred Stock. Thereupon, there shall be issued and delivered to
such holder promptly at such office and in its name as shown on such surrendered
certificate or certificates, a certificate or certificates for the number of
shares of Common Stock into which the shares of Series C Preferred Stock
surrendered were converted on the date on which such automatic conversion
occurred. Such conversion shall be deemed to have been made immediately prior to
the close of business on the Conversion Date, and the person or persons entitled
to receive the shares of Common Stock issuable upon such conversion shall be
treated for all purposes as the record holder or holders of such shares of
Common Stock on such date.

             (c)    Adjustments to Common Stock Conversion Number for Certain
                    ---------------------------------------------------------
Diluting Issues.
---------------
                    (i)     Adjustments for Dividends, Splits, Subdivisions,
                            ------------------------------------------------
Combinations or Consolidation of Common Stock. In the event the outstanding
---------------------------------------------
shares of Common Stock shall be increased by stock dividend payable in Common
Stock, stock split, subdivision, or other similar transaction into a greater
number of shares of Common Stock, the Common Stock Conversion Number of the
Series C Preferred Stock then in effect shall, concurrently with the
effectiveness of such event, be increased proportionally. In the event the
outstanding shares of Common Stock shall be decreased by reverse stock split,
combination, consolidation, or other similar transaction into a lesser number of
shares of Common Stock, the Common Stock Conversion Number of the Series C
Preferred Stock then in effect shall, concurrently with the effectiveness of
such event, be decreased proportionally.

                    (ii)    Adjustments for Other Distributions. In the event
                            -----------------------------------
the Company at any time or from time to time makes, or fixes a record date for
the determination of holders of Common Stock entitled to receive any
distribution payable in securities (other than shares of Common Stock and other
than any distribution which results in the payment of a dividend described in
Section 1), then, and in each such event, provision shall be made so that the
holders of Series C Preferred Stock shall receive upon conversion thereof, in
addition to the number of shares of Common Stock receivable thereupon, the
amount of securities of the Company which they would have received had their
Series C Preferred Stock been converted into Common Stock immediately prior to
the date of such event and had they thereafter, during the period immediately
prior to the date of such event to and including the date of conversion,
retained such securities receivable by them as aforesaid during such period,
subject to all other adjustments called for during such period under this
Section 5 with respect to the rights of the holders of the Series C Preferred
Stock.

                    (iii)  Adjustments for Reclassification, Exchange and
                           ----------------------------------------------
Substitution. If the Common Stock issuable upon conversion of the Series C
------------
Preferred Stock shall be changed into the same or a different number of shares
of any other class or classes of stock, whether by capital reorganization,
reclassification, or otherwise (other than a subdivision or combination of
shares provided for above), then following the effectiveness of such
reorganization or reclassification the Series C Preferred Stock shall be
convertible into, in lieu of the number of shares of Common Stock which the
holders would otherwise have been entitled to receive, a number of shares of
such other class or classes of stock equivalent to the number of shares of such
other class or classes of stock into which the Common Stock was converted in
such reorganization or reclassification that would have been subject to receipt
by the holders upon conversion of such Series C Preferred Stock immediately
before that change.

             (d)    No Impairment.  Except as provided in Section 5, the Company
                    -------------
will not, by amendment of its Certificate of Incorporation or through any
reorganization, transfer of assets, consolidation, merger, dissolution, issue or
sale of securities, or any other voluntary action, avoid or seek to avoid the
observance or performance of any of the terms to be observed or performed
hereunder by the Company but will at all times in good faith assist in the
carrying out of all the provisions of this Section 5

                                      D-3
<PAGE>

and in the taking of all such action as may be necessary or appropriate in order
to protect the conversion rights of the holders of the Series C Preferred Stock
against impairment.
             (e)    Certificate as to Adjustments of the Common Stock Conversion
                    ------------------------------------------------------------
Number. Upon the occurrence of each adjustment or readjustment of the Common
------
Stock Conversion Number for the Series C Preferred Stock pursuant to this
Section 5, the Company at its expense shall promptly compute such adjustment or
readjustment in accordance with the terms hereof and furnish to each holder of
Series C Preferred Stock a certificate setting forth such adjustment or
readjustment and showing in detail the facts upon which such adjustment or
readjustment is based. The Company shall, upon the written request at any time
of any holder of Series C Preferred Stock, furnish or cause to be furnished to
such holder a like certificate setting forth (i) such adjustments and
readjustments, (ii) the Common Stock Conversion Number at the time in effect,
and (iii) the number of shares of Common Stock and the amount, if any, of other
property which at the time would be received upon the conversion of the Series C
Preferred Stock.

             (f)    Notices of Record Date. In the event that this Company shall
                    ----------------------
propose at any time:

                    (i)    to declare any dividend or distribution upon its
Common Stock, whether in cash, property, stock, or other securities, whether or
not a regular cash dividend and whether or not out of earnings or earned
surplus;
                    (ii)   to offer for subscription pro rata to the holders of
any class or series of its stock any additional shares of stock of any class or
series or other rights;
                    (iii)  to effect any reclassification or recapitalization of
its Common Stock outstanding involving a change in the Common Stock; or
                    (iv)   to merge or consolidate with or into any other
corporation, or sell, lease, or convey all or substantially all its property or
business, or to liquidate, dissolve, or wind up; then, in connection with each
such event, this Company shall send to the holders of Series C Preferred Stock:
                           (1)  at least 10 days prior written notice of the
date on which a record shall be taken for such dividend, distribution or
subscription rights (and specifying the date on which the holders of Common
Stock shall be entitled thereto) or for determining rights to vote in respect of
the matters referred to in (iii) and (iv) above; and
                           (2)  in the case of the matters referred to in (iii)
and (iv) above, at least 20 days' prior written notice of the date when the same
shall take place (and specifying the date on which the holders of Common Stock
shall be entitled to exchange their Common Stock for securities or other
property deliverable upon the occurrence of such event or the record date for
the determination of such holders if such record date is earlier).

     Each such written notice shall be delivered personally, or given by first
class mail, postage prepaid, addressed to the holders of the Series C Preferred
Stock at the address for each such holder as shown on the books of this Company.
             (g)    Issue Taxes.  The Company shall pay any and all issue and
                    -----------
other taxes (other than income taxes) that may be payable in respect of any
issue or delivery of shares of Common Stock on conversion of shares of Series C
Preferred Stock pursuant hereto; provided, however, that the Company shall not
be obligated to pay any transfer taxes resulting from any transfer requested by
any holder in connection with any such conversion.
             (h)    Reservation of Stock Issuable Upon Conversion. At the
                    ---------------------------------------------
Conversion Date, the Company shall have reserved and have available out of its
authorized but unissued shares of Common Stock, solely for the purpose of
effecting the conversion of the shares of the Series C Preferred Stock such
number of its shares of Common Stock as shall be sufficient to effect the
conversion of all outstanding shares of Series C Preferred Stock; and if at any
time the number of authorized but unissued shares of Common Stock shall not be
sufficient to effect the conversion of all then outstanding shares of the Series
C Preferred Stock, the Company will take such corporate action as may, in the
opinion of its counsel, be necessary to increase its authorized but unissued
shares of Common Stock to such number of shares as shall be sufficient for such
purpose, including, without limitation, engaging in best efforts to obtain the
requisite stockholder approval of any necessary amendment to its Certificate of
Incorporation.

                                      D-4
<PAGE>

             (i)    Defined Terms
                    -------------
             As used herein, the following terms as have the meanings set forth
below:

             "Conversion Date," as applied to any share of Series C Preferred
Stock, shall mean the date on which the Company's right of repurchase lapses
with respect to such share of Series C Preferred Stock pursuant to the Amendment
to Stock Purchase Agreement or Amendment to Notice of Stock Option Grant and
Stock Option Agreement between the Company and each of the holders of shares of
Series C Preferred Stock.
             "Common Stock Conversion Number" shall mean one thousand (1,000)
shares of Common Stock, as adjusted pursuant to Section 5(c) hereof.

         6.  Consolidation, Merger, etc. In case the Company shall enter into
             --------------------------
any consolidation, merger, combination or other transaction in which the shares
of Common Stock are exchanged for or changed into other stock or securities,
cash and/or other property, then in any such case each share of Series C
Preferred Stock shall at the same time be similarly exchanged or changed into an
amount per share, subject to the provision for adjustment hereinafter set forth,
equal to 1,000 times the aggregate amount of stock, securities, cash and/or any
other property (payable in kind), as the case may be, into which or for which
each share of Common Stock is changed or exchanged. In the event the Company
shall at any time declare or pay any dividend on the Common Stock payable in
shares of Common Stock, or effect a subdivision or combination or consolidation
of the outstanding shares of Common Stock (by reclassification or otherwise than
by payment of a dividend in shares of Common Stock) into a greater or lesser
number of shares of Common Stock, then in each such case the amount set forth in
the preceding sentence with respect to the exchange or change of shares of
Series C Preferred Stock shall be adjusted by multiplying such amount by a
fraction, the numerator of which is the number of shares of Common Stock
outstanding immediately after such event and the denominator of which is the
number of shares of Common Stock that were outstanding immediately prior to such
event.

         7.  No Redemption.  The shares of Series C Preferred Stock shall not be
             -------------
redeemable.

     IN WITNESS WHEREOF, this Certificate of Designation is executed on behalf
of the Company this 9th day of February, 2000.

                                            AT HOME CORPORATION

                                            By:  /s/ David Pine
                                               --------------------------
                                               David Pine, Secretary

                                      D-5

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