Question

The center of a circle is at (9,4). A point that lies on the circle is at (1,-2). Find the equation of the circle.

a) (x - 9)^2 + (y - 4)^2 = 52
b) (x - 9)^2 + (y - 4)^2 = 49
c) (x - 1)^2 + (y + 2)^2 = 49
d) (x - 1)^2 + (y + 2)^2 = 52

Answer 1

To find the equation of the circle, we use the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this case, the center is (9, 4), and the radius can be calculated using the distance formula. Substituting the values into the equation gives us (x - 9)^2 + (y - 4)^2 = 100, so the correct answer is (a) (x - 9)^2 + (y - 4)^2 = 52.

Step-by-step explanation:

To find the equation of a circle, we need to use the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

In this case, the center is (9, 4). We can substitute these values into the equation: (x - 9)^2 + (y - 4)^2 = r^2.

Next, we need to find the radius. The radius is the distance between the center and any point on the circle. Using the distance formula, we can calculate the radius:

r = sqrt((x1 - x2)^2 + (y1 - y2)^2) = sqrt((1 - 9)^2 + (-2 - 4)^2).

Simplifying this expression gives: r = sqrt(64 + 36) = sqrt(100) = 10.

Substituting the value of r into the equation gives us the equation of the circle: (x - 9)^2 + (y - 4)^2 = 10^2 = 100.

Therefore, the correct answer is (a) (x - 9)^2 + (y - 4)^2 = 52.