Answer
$x^{2}+y^{2}=65$
Work Step by Step
Center at the origin; passes through $(4,7)$
The equation of a circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $(h,k)$ is the center of the circle and $r$ is its radius.
The center of the circle and a point through which it passes are given.
Use the formula for the distance between two points to obtain the distance between the center of the circle and the point given. This distance represents the radius of the circle. The formula is $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
$d=\sqrt{(4-0)^{2}+(7-0)^{2}}=\sqrt{16+49}=\sqrt{65}$
Since now the center and the radius of the circle are known, substitute them into the formula for the equation of a circle:
$(x-h)^{2}+(y-k)^{2}=r^{2}$
$(x-0)^{2}+(y-0)^{2}=(\sqrt{65})^{2}$
$x^{2}+y^{2}=65$
