Question

If the endpoints of the diameter of a circle are (8, −6) and (4, −2), what is the standard form equation of the circle?

Answer 1

We do the midpoint formula to find the center of the circle to get the left side of the equation.

Midpoint = [(X₁ + X₂) / 2 , (Y₁ + Y₂) / 2] 

Now plug in:

[(8 + 4) / 2 , (- 6 - 2) / 2]
(12 / 2 , - 8 / 2)
(6, - 4)

The center of the circle is (6, -4)

Now we plug it into the equation of a circle:

(x - h)² + (y - k)² = r² where (h, k) is the center of the circle and r is the radius. 

(x - 6)² + (y + 4)² = r² is the left side of the equation. This will eliminate options A and C

Now we do the distance formula using the center and an endpoint to get the radius. The formula for distance is:

√((X₂ - X₁)² + (Y₂ - Y₁)²)

We plug in using either of the endpoints. 

√((4 - 6)² + (- 2 - (- 4))²)
√((-2)² + 2²)
√(4 + 4)
√8

√8 is your radius

(√8)² = 8

Your correct answer is (x - 6)² + (y + 4)² = 8, Option D