# Famous Theorems of Mathematics/Geometry/Conic Sections
## Parabola Properties

Prove for point (x,y) on a parabola with focus (h,k+p) and directrix y=k-p, that:
$(x-h)^{2}=4p(y-k)$
and that the vertex of this parabola is (h,k)
| Statement | Reason |
| (1) Arbitrary real value h | Given |
| (2) Arbitrary real value k | Given |
| (3) Arbitrary real value p where p is not equal to 0 | Given |
| (4) Line l, which is represented by the equation $y=k-p$ | Given |
| (5) Focus F, which is located at $(h,k+p)$ | Given |
| (6) A parabola with directrix of line l and focus F | Given |
| (7) Point on parabola located at $(x,y)$ | Given |
| (8) Point (x, y) must is equidistant from point f and line l. | Definition of parabola |
| (9) The distance from (x, y) to l is the length of line segment which is both perpendicular to l and has one endpoint $P_{1}$ on l and one endpoint $P_{2}$ on (x, y). | Definition of the distance from a point to a line |
| (10) Because the slope of l is 0, it is a horizontal line. | Definition of a horizontal line |
| (11) Any line perpendicular to l is vertical. | If a line is perpendicular to a horizontal line, then it is vertical. |
| (12) All points contained in a line perpendicular to l have the same x-value. | Definition of a vertical line |
| (13) Point $P_{1}$ has a y-value of $k-p$. | (4) and (9) |
| (14) Point $P_{1}$ has an x-value of x. | (7), (9), and (12) |
| (15) Point $P_{1}$ is located at (x, k - p). | (13) and (14) |
| (16) Point $P_{2}$ is located at (x, y). | (9) |
| (17) $P_{1}P_{2}={\sqrt {(x-x)^{2}+(y-[k-p])^{2}}}$ | Distance Formula |
| (18) $P_{1}P_{2}={\sqrt {(y-k+p)^{2}}}$ | Distributive Property |
| (19) $P_{1}P_{2}=(y-k+p)$ | Apply square root; distance is positive |
| (20) $FP_{2}={\sqrt {(x-h)^{2}+(y-[k+p])^{2}}}$ | Distance Formula |
| (21) $FP_{2}={\sqrt {(x-h)^{2}+(y-k-p)^{2}}}$ | Distributive Property |
| (22) $FP_{2}=P_{1}P_{2}$ | Definition of Parabola |
| (23) ${\sqrt {(x-h)^{2}+(y-k-p)^{2}}}=(y-k+p)$ | Substitution |
| (24) $(x-h)^{2}+(y-k-p)^{2}=(y-k+p)^{2}$ | Square both sides |
| (25) $(x-h)^{2}+k^{2}+p^{2}+y^{2}+2kp-2ky-2py=k^{2}+p^{2}+y^{2}-2kp-2ky+2py$ | Distributive property |
| (26) $(x-h)^{2}+2kp-2py=2py-2kp$ | Subtraction Property of Equality |
| (27) $(x-h)^{2}=4py-4kp$ | Addition Property of Equality; Subtraction Property of Equality |
| (28) $(x-h)^{2}=4p(y-k)$ | Distributive Property |

### Finding the Axis of Symmetry

| Statement | Reason |
| (29) The axis of symmetry is vertical. | (10); Definition of axis of symmetry; if a line is perpendicular to a horizontal line, then it is vertical |
| (30) The axis of symmetry contains (h, k + p). | Definition of Axis of Symmetry |
| (31) All points in the axis of symmetry have an x-value of h. | Definition of a vertical line; (30) |
| (32) The equation for the axis of symmetry is $x=h$. | (31) |

### Finding the Vertex

| Statement | Reason |
| (33) The vertex lies on the axis of symmetry. | Definition of the vertex of a parabola |
| (34) The x-value of the vertex is h. | (33) and (32) |
| (35) The vertex is contained by the parabola. | Definition of vertex |
| (36) $(h-h)^{2}=4p(y-k)$ | (35); Substitution: (28) and (34) |
| (37) $0=4p(y-k)$ | Simplify |
| (38) $0=y-k$ | Division Property of Equality |
| (39) $k=y$ | Addition Property of Equality |
| (40) $y=k$ | Symmetrical Property of Equality |
| (41) The vertex is located at $(h,k)$. | (34) and (40) |