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?

(x - 2)^2 + (y + 10)^2 = 13
(x - 2)^2 + (y + 10)^2 = 169
(x + 2)^2 + (y - 10)^2 = 13
(x + 2)^2 + (y - 10)^2 = 169

Explanation:

⚡ Using what you learned: Equations of Circles

Step 1: Identify the center

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Given the center \((h, k) = (-2, 10)\):

$$ (x - (-2))^2 + (y - 10)^2 = r^2 $$

$$ (x + 2)^2 + (y - 10)^2 = r^2 $$

Step 2: Calculate the radius squared

Substitute the point \(P(10, 5)\) into the equation to find \(r^2\):

$$ (10 + 2)^2 + (5 - 10)^2 = r^2 $$

$$ (12)^2 + (-5)^2 = r^2 $$

$$ 144 + 25 = r^2 $$

$$ r^2 = 169 $$

Step 3: Write the final equation

Substitute \(r^2 = 169\) back into the equation:

$$ (x + 2)^2 + (y - 10)^2 = 169 $$

Answer:

$$ (x + 2)^2 + (y - 10)^2 = 169 $$