Question

Determine the equation of the circle with center (8,5) containing the point (-4,21)

Answer 1

(x-8)² + (y-5)² = 400

Explanation:

general equation of a circle : (x-h)² + (y-k)² = r²

where (h,k) is at center and r = radius

here we are given the center as well as a point on the circle

first we will need to identify the radius by finding the distance between the center and the point on the circle using using the distance formula ( we know this is the radius because the radius is a line that goes on the center of the circle to any point on the circle )

Identifying the radius using the distance formula

Distance formula :
`d=√((x_2-x1)^2+(y_2-y_1)^2)`

where the x and y values are derived from the two given points

Given points are (8,5) and (-4,21)

Assigning variables we get (x1,y1) = (8,5) , so x1 = 8 and y1 = 5

and (x2,y2) = (-4,21) so x2 = -4 and y2 = 21

Plugging this into the formula we get

`d=√((-4-8)^2+(21-5)^2)`

==> subtract values inside of parenthesis

`d=√((-12)^2+(16)^2)`

==> evaluate exponents

`d = √(144+256)`

==> add 144 and 256

`d=√(400)`

==> take the square root of 400

`d=20`

so the radius is 400

Finding the equation of the circle

again we have the general equation of a circle as (x-h)² + (y-k)² = r²

where (h,k) is at center and r = radius

here the center is at (8,5) and the radius is 20

so (h,k) = (8,5) so h = 8 and k = 5 and radius is 20 so r = 20

plugging these values into the general equation of a circle we get

(x-8)² + (y-5)² = 20²

==> evaluate exponent

(x-8)² + (y-5)² = 400

and we are done!

For more validation check the attached image :)