Answer:
(x - 7)² + (y - 5)² = 78
Step-by-step 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 = [tex]\sqrt{(3-7)^2+(13-5)^2}[/tex]
= [tex]\sqrt{(-4)^2+8^2}[/tex]
= [tex]\sqrt{16+64}[/tex] = [tex]\sqrt{78}[/tex]
Hence
(x - 7)² + (y - 5)² = ([tex]\sqrt{78}[/tex])², that is
(x - 7)² + (y - 5)² = 78 ← equation of circle
