Question

determine the equation of the circle if its center is (8,-6) and which passes through the points (5,-2).

Answer 1

(x - 8)² + (y + 6)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 ) = (8, - 6 ) , then

(x - 8)² + (y - (- 6))² = r² , that is

(x - 8)² + (y + 6)² = r²

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

Calculate r using the distance formula

r =
`\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2 }`

with (x₁, y₁ ) = (8, - 6 ) and (x₂, y₂ ) = (5, - 2 )

r =
`√((5-8)^2+(-2-(-6))^2)`

=
`√((-3)^2+(-2+6)^2)`

=
`√(9+4^2)`

=
`√(9+16)`

=
`√(25)`

= 5

Then equation of circle is

(x - 8)² + (y + 6)² = 5² , that is

(x - 8)² + (y + 6)² = 25