SAT Question d12bca0a: Circles answer and explanation with Desmos solution

Choice D is correct. It’s given that the circle shown has its center at $(0,0)$. The equation of a circle in the xy-plane with its center at $(0,0)$ can be expressed as $x^2+y^2=r^2$, where $r$ is the radius of the circle. It’s shown that $(8,0)$ lies on the circle. Since the circle has its center at $(0,0)$ and $(8,0)$ lies on the circle, it follows that the radius of the circle is $8$. Substituting $8$ for $r$ in the equation $x^2+y^2=r^2$ yields $x^2+y^2=64$. It's also shown that $(2,k)$ lies on the circle. Substituting $2$ for $x$ and $k$ for $y$ in this equation yields $2^2+k^2=64$, or $4+k^2=64$. Subtracting $4$ from both sides of this equation yields $k^2=60$. Taking the square root of both sides of this equation yields $k=\pm \sqrt{60}$. Since $(2,k)$ lies above the x-axis, $k$ must be positive. Therefore $k=\sqrt{60}$.