Mathematics: The endpoints of a diameter of a circle are located at (5,9) and (11,17) which is an equation of the circle?

The endpoints of a diameter of a circle are located

chuchi24

10.03.2022

(x - 8)² + (y - 13)² = 25

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 located at the midpoint of the endpoints of the diameter.

Use the midpoint formula to find the centre

[ $\frac{x_{1}+x_{2} }{2}$ , $\frac{y_{1}+y_{2} }{2}$ ]

with (x₁, y₁ ) = (5, 9) and (x₂, y₂ ) = (11,17)

centre = ( $\frac{5+11}{2}$ , $\frac{9+17}{2}$ ) = (8, 13)

The radius is the distance from the centre to either end of the diameter

Calculate r using the distance formula

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (8, 13) and (x₂, y₂ ) = (5, 9)

r = $\sqrt{(5-8)^2+(9-13)^2}$

= $\sqrt{(-3)^2+(-4)^2}$

= $\sqrt{9+16}$ = $\sqrt{25}$ = 5 ⇒ r² = 25

Hence

(x - 8)² + (y - 13)² = 25

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