Question

Write the standard form of the equation of the circle which the given characteristics endpoints of a diameter (11, -3), (3, 13)

Answer 1

(x - 7)² + (y - 5)² = 78

Explanation:

The standard form of the equation of a circle 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

Calculate the centre using the midpoint formula

midpoint = [0.5(x₁ + x₂), 0.5(y₁ + y₂) ]

with (x₁, y₁ ) = (11, - 3) and (x₂, y₂ ) = (3, 13)

centre = [0.5(11 + 3), 0.5(- 3 + 13) ] = (7, 5)

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

Calculate the radius using the distance formula

d = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (7, 5) and (x₂, y₂ ) = (3, 13)

r =
`√((3-7)^2+(13-5)^2)`

=
`√((-4)^2+8^2)`

=
`√(16+64)` =
`√(78)`

Hence

(x - 7)² + (y - 5)² = (
`√(78)`)², that is

(x - 7)² + (y - 5)² = 78 ← equation of circle