Answer
$r=-14 \sin\theta$
.
Work Step by Step
The circle passes through the origin, as (0,0) satisfies the Cartesian equation.
The radius is $7$, and the center is at $(0,-7)$.
In polar coordinates, the center lies at $r_{0}=7, \theta_{0}=3\pi/2,\qquad P(7,-\pi/2)$
A circle passing through the origin, of radius $a$, centered at $P_{0}(r_{0}, \theta_{0}),$
has the polar equation$ \quad r=2a\cos(\theta-\theta_{0})$
So this circle has the equation
$r=2(7)\displaystyle \cos(\theta+\frac{\pi}{2})$
or
$r=14\displaystyle \cos(\theta+\frac{\pi}{2})$
Applying the trigonometric identity
$\displaystyle \cos(\theta\pm\frac{\pi}{2})=\mp\sin\theta$,
we can rewrite the equation as
$r=-14 \sin\theta$
