Question

How do you find the center and radius of the circle `4x^2 - 16x + 4y^2 - 24y + 36 = 0`?

Answer 1

`4x^2-16x+4y^2-24y+36=0` is a circle with center at `(2,3)` and radius `2`.

Explanation:

Dividing each term by `4` in `4x^2-16x+4y^2-24y+36=0`, we get

`x^2-4x+y^2-6y+9=0`

Now for converting `x^2-4x` and `y^2-6y` into complete squares, we need to add `4` and `9` respectively. As `9` is already there, adding `4` on both sides, we get

`x^2-4x+4+y^2-6y+9=4`

or `(x-2)^2+(y-3)^2=2^2`

This is the equation of a circle with center at `(2,3)` and radius `2`. Its graph is as given below.
graph{4x^2-16x+4y^2-24y+36=0 [-8.46, 11.54, -2.28, 7.72]}