Question

The endpoints of a diameter of a circle are located at (5,9) and (11,17) which is an equation of the circle?

Answer 1

(x - 8)² + (y - 13)² = 25

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 located at the midpoint of the endpoints of the diameter.

Use the midpoint formula to find the centre

[
`(x_(1)+x_(2)  )/(2)`,
`(y_(1)+y_(2)  )/(2)` ]

with (x₁, y₁ ) = (5, 9) and (x₂, y₂ ) = (11,17)

centre = (
`(5+11)/(2)`,
`(9+17)/(2)` ) = (8, 13)

The radius is the distance from the centre to either end of the diameter

Calculate r using the distance formula

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (8, 13) and (x₂, y₂ ) = (5, 9)

r =
`√((5-8)^2+(9-13)^2)`

=
`√((-3)^2+(-4)^2)`

=
`√(9+16)` =
`√(25)` = 5 ⇒ r² = 25

Hence

(x - 8)² + (y - 13)² = 25