Question

circles a and b are graphed in the xy - plane. circle a is represented by the equation (x + 8)^2+(y - 8)^2 = 64 and intersects the x - axis at the point (r,s). circle b has its center at (r,s) and has a radius of the same length as circle a. which equation represents circle b? a. x^2+(y + 8)^2 = 64 b. x^2+(y - 8)^2 = 64 c. (x + 8)^2+y^2 = 64 d. (x - 8)^2+y^2 = 64

Explanation:

Step1: Recall circle - equation formula

The standard form of a circle's equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius. The equation of circle \(A\) is \((x + 8)^{2}+(y - 8)^{2}=64\), so the center of circle \(A\) is \((-8,8)\) and the radius \(r = 8\) (since \(r^{2}=64\)).

Step2: Find the center of circle \(A\)'s intersection with the \(x\) - axis

Set \(y = 0\) in the equation of circle \(A\): \((x + 8)^{2}+(0 - 8)^{2}=64\), \((x + 8)^{2}+64=64\), \((x + 8)^{2}=0\), \(x=-8\). So the circle \(A\) intersects the \(x\) - axis at the point \((-8,0)\), then \(r=-8\) and \(s = 0\), and the center of circle \(B\) is \((-8,0)\).

Step3: Write the equation of circle \(B\)

Since the center of circle \(B\) is \((-8,0)\) and the radius is the same as circle \(A\) (\(r = 8\)), using the standard - form of the circle equation \((x - h)^{2}+(y - k)^{2}=r^{2}\), substituting \(h=-8\), \(k = 0\), and \(r = 8\) into it, we get \((x+8)^{2}+y^{2}=64\).

Answer:

C. \((x + 8)^{2}+y^{2}=64\)