college algebra

Question

find the equation for the circle with a diameter whose endpoints are

​ (−5​,−3​)

and (1,2)

Answer 1

An equation of the circle is (x-h)² + (y-k)² = f², where (h,k) is the center and r is the radius.

The center is the midpoint of any diameter and r is half the length of any diameter.

Answer 2

find the equation for the circle with a diameter whose endpoints are (−5​,−3​) and (1,2)

Detailed Solution:

Equation of the circle: (x - h)² + (y - k)² = r²

Endpoints: (x₁, y₁) =(−5​,−3​) (x₂, y₂) = (1, 2)

Midpoint - Center of Circle: (h, k) = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((−5 + 1)/2, (−3 + 2)/2) = (−4/2, −1/2) = (−2, −1/2)

Diameter: d = √(x₂ − x₁)² + ( y₂ − y₁)² = √(1 − (−5))² + ( 2 −(−3))² = √(6)² + (5)² = √(36+25) = √(61)

r = d/2 ==> r = √(61)/2

Midpoint - Center of Circle: (h, k) = (−2, −1/2) r = √(61)/2

Equation of the circle: (x - h)² + (y - k)² = r² ==> (x − (−2))² + (y − (−1/2))² = (√(61)/2)²

(x + 2)² + (y + 1/2)² = 61/4