Question

The endpoints of the diameter of a circle are (6, 5) and (−2, 3). Which equation represents the circle?

Answer 1

D) (x − 2)2 + (y − 4)2 = 17

Explanation:

The midpoint of the diameter is the center of the circle.

Endpoints of diameter: (6, 5) and (−2, 3)

6 −2/ 2 = 2; 5 + 3/2 = 4 → midpoint: (2, 4)

r = distance between center and either end point of the diameter

Distance formula: d = (x2 - x1)2 + (y2 - y1)2

center: (2, 4); point: (6, 5)

r = (6 − 2)2 + (5 − 4)2 → 16 + 1 → 17

(x − h)2 + (y − k)2 = r2 → (x − 2)2 + (y − 4)2 = 17

Answer 2

(x - 2)² + (y - 4)² = 17

Explanation:

The equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k) are the coordinates of the centre and r is the radius

The centre is at the midpoint of the endpoints of the diameter, that is

x =
`(6-2)/(2)` =
`(4)/(2)` = 2 and y =
`(5+3)/(2)` =
`(8)/(2)` = 4

centre = (2, 4 )

The radius is the distance from the centre to either of the endpoints

Using the distance formula to find r

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (2, 4) and (x₂, y₂ ) = (- 2, 3)

r =
`√((-2-2)^2+(3-4)^2)`

=
`√((-4)^2+(-1)^2)`

=
`√(16+1)`

=
`√(17)` ⇒ r² = 17

Thus the equation of the circle is

(x - 2)² + (y - 4)² = 17