Determine the equation of a circle in standard form with endpoints at (-3, 5) and (7, 1) then state the center and radius

(x - 2)² + (y - 3)² = 29

Step-by-step 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 at the midpoint of the endpoints of the diameter.

Use the midpoint formula to find the centre

[ 0.5(x₁ + x₂ ), 0.5(y₁ + y₂ ) ]

with (x₁, y₁ ) = (- 3, 5) and (x₂, y₂ ) = (7, 1), thus

centre = [ 0.5(- 3+7), 0.5(5 + 1) ] = [0.5(4), 0.5(6) ] = (2. 3 )

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

Calculate the radius using the distance formula

√ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (2, 3) and (x₂, y₂ ) = (7, 1)

r =

  =

  = = , hence

(x - 2)² + (y - 3)² = ( )², that is

(x - 2)² + (y - 3)² = 29 ← equation of circle

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