Write the equation of the circle whose diameter has endpoints (-1,2)(3,-2)

Answer 1

The equation of a circle is (x-h)² + (y-k)² = r² where the center is (h,k) and the r represents the radius. The diameter can help us find both of those things. A diameter is a line passes from side to side going through the center. You can think of it as a line that would cut the circle in half.

Because the diameter passes from side to side it means that the end points are on each side of the circle. This means that midpoint of the diameter would be the center of the circle. To find a midpoint we want to take the average of the x's and the average of the y's like so,

(x₁ + x₂)/2 , (y₁+ y₂)/2

For this problem it would look like this:

endpoint 1: (-1, 2) endpoint 2: (3,-2)

(x_1, y₁) (x₂, y₂)

(-1 + 3)/2 , (2 + -2)/2

both x both y

2/2 , 0/2

(1 , 0) : center!

(h , k)

We can plug that into our formula: (x-1)² + (y-0)² = r².

Now we need to find the radius. The radius is half of the diameter, so we need to first find the length of the diameter then divide it by 2.

distance = √((x₁-x₂)² + (y₁-y₂)² = √((-1 - 3)² + (2 - (-2))²) = √((-4)²+ (4)²) = √(16 + 16) = √32 = 4√2

Diameter length is 4√2. Radius = d/2 = (4√2)/2 = 2√2 . The radius length is 2√2.

All together, (x-1)² + (y-0)² = (2√2)²

(x-1)² + y² = 8