SAT Question 981275d2: Circles answer and explanation with Desmos solution

Choice A is correct. The standard form for the equation of a circle is $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ are the coordinates of the center and $r$ is the length of the radius. According to the given equation, the center of the circle is $(6,-5)$. Let $(x_1,y_1)$ represent the coordinates of point $Q$. Since point $P$ $(10,-5)$ and point $Q$ $(x_1,y_1)$ are the endpoints of a diameter of the circle, the center $(6,-5)$ lies on the diameter, halfway between $P$ and $Q$. Therefore, the following relationships hold: $\frac{x_1+10}{2}=6$ and $\frac{y_1+(-5)}{2}=-5$. Solving the equations for $x_1$ and $y_1$, respectively, yields $x_1=2$ and $y_1=-5$. Therefore, the coordinates of point $Q$ are $(2,-5)$.

Alternate approach: Since point $P$ $(10,-5)$ on the circle and the center of the circle $(6,-5)$ have the same $y$-coordinate, it follows that the radius of the circle is $10-6=4$. In addition, the opposite end of the diameter $PQ$ must have the same $y$-coordinate as $P$ and be $4$ units away from the center. Hence, the coordinates of point $Q$ must be $(2,-5)$.