Question

Determine the equation of a circle whose diameter has the endpoints (-1, 2) and (7.

-4).
1) (x - 3)2 + (y + 1)2 = 5
2) (x - 3)2 + (y + 1)2 = 25
3) (x - 3)2 + (y + 1)2 = 100
4) (x - 3)2 + (y + 1)2 = 10

Answer 1

(x - 3)^2 + (y + 1) = 25

Explanation:

First find the midpoint of the diameter, because that represents the center of the circle.

-1 + 7

The x-coordinate of the midpoint is xm = ---------- = 3

2

2 - 4

and the y-coordinate is ym = ---------- = -1

2

And so the center of this circle is at (3, -1).

Use the Pythagorean Theorem to determine the square of the radius:

square of radius = 4^2 + (-3)^2 = 16 + 9 = 25

And so the equation of this circle is (x - 3)^2 + (y + 1) = 25

Answer 2

option 2

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.

Using the midpoint formula with (- 1, 2) and (7, - 4), then

centre = (
`(-1+7)/(2)` ,
`(2-4)/(2)` ) = (3, - 1 )

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

Using the distance formula

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

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

r =
`√((3+1)^2+(2+1)^2)`

=
`√(4^2+3^2)`

=
`√(16+9)` =
`√(25)` = 5

Thus

(x - 3)² + (y - (- 1))² = 5² , that is

(x - 3)² + (y + 1)² = 25 ← equation of circle