Answer
$r=10 \sin\theta$
.
Work Step by Step
The circle passes through the origin, as (0,0) satisfies the Cartesian equation.
The radius is $5$, and the center is at $(0,5)$.
In polar coordinates, the center lies at $r_{0}=5, \theta_{0}=\pi/2,\qquad P(5,\pi/2)$
A circle passing through the origin, of radius $a$, centered at $P_{0}(r_{0}, \theta_{0}),$
has the polar equation
$r=2a\cos(\theta-\theta_{0})$
So this circle has the equation
$r=2(5)\displaystyle \cos(\theta-\frac{\pi}{2})$
or
$r=10\displaystyle \cos(\theta-\frac{\pi}{2})$
Applying the trigonometric identity
$\displaystyle \cos(\theta\pm\frac{\pi}{2})=\mp\sin\theta$,
we can rewrite the equation as
$r=10 \sin\theta$
