Question

which of the following is the equation for a circle with a radius of r and center at (h, v)?
a. (x + h)^2 + (y + v)^2 = r^2
b. h^2 + v^2 = r^2
c. (x - h)^2 + (y - v)^2 = r^2
d. (x - v)^2 + (y - h)^2 = r^2

Explanation:

Step1: Recall circle - equation formula

The standard - form equation of a circle with center \((h,k)\) (in this case \((h,v)\)) and radius \(r\) is \((x - h)^2+(y - v)^2=r^2\). This formula is derived from the distance formula. The distance between a point \((x,y)\) on the circle and the center \((h,v)\) is equal to the radius \(r\). By the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \((x_1,y_1)=(h,v)\) and \((x_2,y_2)=(x,y)\) and \(d = r\), we get \(r=\sqrt{(x - h)^2+(y - v)^2}\), and squaring both sides gives \((x - h)^2+(y - v)^2=r^2\).

Answer:

C. \((x - h)^2+(y - v)^2=r^2\)