Question

which equation represents a circle centered at (3,5) and passing through the point (-2,9)?
a. ((x - 3)^2 + (y - 5)^2 = 17)
b. ((x - 3)^2 + (y - 5)^2 = 41)
c. ((x + 3)^2 + (y + 5)^2 = 41)
d. ((x + 3)^2 + (y + 5)^2 = 17)

Explanation:

Step1: Recall the 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 and \(r\) is the radius. Here, the center \((h,k)=(3,5)\), so the equation starts as \((x - 3)^2 + (y - 5)^2 = r^2\). This eliminates options C and D since their centers are \((-3,-5)\), not \((3,5)\).

Step2: Calculate the radius squared

The radius is the distance from the center \((3,5)\) to the point \((-2,9)\) on the circle. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so \(r^2=(x_2 - x_1)^2+(y_2 - y_1)^2\). Substituting \(x_1 = 3,y_1 = 5,x_2=-2,y_2 = 9\):
\(r^2=(-2 - 3)^2+(9 - 5)^2=(-5)^2+(4)^2 = 25 + 16 = 41\).

Answer:

B. \((x - 3)^2 + (y - 5)^2 = 41\)