options: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Derive the equation of a parabola given a focus and directrix.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
problem_activity: Task
This problem examines equations defining different circles in the $x$-$y$ plane.
Use the Pythagorean theorem to
find an equation in $x$ and $y$ whose solutions are the points on the circle  of radius 2 with center $(1,1)$  and explain why it works.
Suppose $r$ is a positive number and $(a,b)$ a point in the plane.
Use the Pythagorean theorem to
find an equation in $x$ and $y$ whose solutions are the points on the circle of radius $r$ with center $(a,b)$ and explain why it works.


correct_option_index: 2