Question

Write the equation of a circle with a center at (1,4) where a point on the circle is (4,8).

Answer 1

(x - 1)² + (y - 4)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

here (h, k) = (1, 4), so

(x - 1)² + (y - 4)² = r²

The radius is the distance from the centre to a point on the circle

Calculate r using the distance formula

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (1, 4) and (x₂, y₂ ) = (4, 8)

r =
`√((4-1)^2+(8-4)^2)`

=
`√(3^2+4^2)` =
`√(9+16)` =
`√(25)` = 5

Hence

(x - 1)² + (y - 4)² = 5², that is

(x - 1)² + (y - 4)² = 25 ← equation of circle