Question

select the correct answer. circle c has a center at (-2,10) and contains the point p(10,5). which equation represents circle c? a. (x - 2)² + (y + 10)² = 13 b. (x - 2)² + (y + 10)² = 169 c. (x + 2)² + (y - 10)² = 13 d. (x + 2)² + (y - 10)² = 169

Explanation:

Step1: Recall the circle equation

The standard equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Identify the center coordinates

Given the center of circle \(C\) is \((-2,10)\), so \(h=-2\) and \(k = 10\). Substituting these into the equation, we get \((x - (-2))^2+(y - 10)^2=r^2\), which simplifies to \((x + 2)^2+(y - 10)^2=r^2\). This eliminates options A and B.

Step3: Calculate the radius

The circle contains the point \(P(10,5)\). The radius \(r\) is the distance between the center \((-2,10)\) and the point \(P(10,5)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(x_1=-2,y_1 = 10,x_2=10,y_2=5\). So \(r=\sqrt{(10 - (-2))^2+(5 - 10)^2}=\sqrt{(12)^2+(-5)^2}=\sqrt{144 + 25}=\sqrt{169}=13\). Then \(r^2=169\).

Answer:

D. \((x + 2)^2+(y - 10)^2 = 169\)