Equation of circle with given radius passing through two given points

Question

Find the equation of the circles passing through two points on the $y$ axis at distances 3 units from the origin and having radius 5.

(This a homework problem but I do not know how to solve it.)

Answer 1

You are looking for the circle which intersects the points $(0, 3)$ and $(0, -3)$, whose radius is $5$. To find the center of the circle, you need to find the point $(x_0, y_0)$ such that $$\sqrt{(x_0 - 0)^2 + (y_0 - 3)^2} = \sqrt{(x_0 - 0)^2 + (y_0 + 3)^2} = 5$$

Note that this center must necessarily be located on the $x$-axis: $(x_0, y_0) = (x_0, 0)$, determined by $x_0, -x_0$, each of which will define a distinct circle.

The general equation of a circle with center $(x_0, y_0)$ and radius $r$ is given by:

$$(x - x_0)^2 + (y - y_0)^2 = r^2$$