Question

What is the standard equation of a circle with center (0, 2) that
passes through (5,-2)?

Answer 1

x² + (y - 2)² = 41

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

calculate r using the distance formula

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

with (x₁, y₁ ) = (0, 2 ) and (x₂, y₂ ) = (5, - 2 )

r =
`√((5-0)^2+(-2-2)^2)`

=
`√(5^2+(-4)^2)`

=
`√(25+16)`

=
`√(41)` , then r² = (
`√(41)` )² = 41

then

(x - 0)² + (y - 2)² = 41 , that is

x² + (y - 2)² = 41 ← equation of circle