Question

What is the center and radius of the circle described by this equation `x^ { 2} - 16x + y ^ { 2} - 12y = - 51`?

Answer 1

This is the equation of a circle, center `(8,6)` and the radius is `=7`

Explanation:

We need

`a^2-2ab+b^2=(a-b)^2`

The equation of a circle, center `(a,b)` and radius `=r` is

`(x-a)^2+(y-b)^2=r^2`

Our equation is

`x^2-16x+y^2-12y=-51`

We complete the squares by adding half the coefficient of `x` and `y` to the square on both sides of the equation

`x^2-16+64+y^2-12y+36=-51+64+36`

We factorise the LHS

`(x-8)^2+(y-6)^2=49=7^2`

This is the equation of a circle, center `(8,6)` and the radius is `=7`

graph{(x^2-16x+y^2-12y+51)=0 [-7.42, 21.06, -1.37, 12.88]}