Question

Complete the square to rewrite the following equation. Identify the center and radius of the circle. You must show all work and calculations to receive credit. x2 + 4x + y2 − 6y = −4

Answer 1

Given:

The equation of a circle is:

$x^2+4x+y^2-6y=-4$

To find:

The center and radius of the circle.

Solution:

We have,

$x^2+4x+y^2-6y=-4$

To make complete square add square of half of x and square of half of y on both sides.

$x^2+4x+y^2-6y+\left(\dfrac{4}{2}\right)^2+\left(\dfrac{-6}{2}\right)^2=-4+\left(\dfrac{4}{2}\right)^2+\left(\dfrac{-6}{2}\right)^2$

$x^2+4x+y^2-6y+\left(2\right)^2+\left(-3\right)^2=-4+(2)^2+(-3)^2$

$(x^2+4x+4)+(y^2-6y+9)=-4+4+9$

$(x+2)^2+(y-3)^3=9$

$(x+2)^2+(y-3)^3=3^2$             ...(i)

The standard form of a circle is:

$(x-h)^2+(y-k)^2=r^2$              ...(ii)

On comparing both sides, we get

$h=-2,k=3,r=3$

Therefore, the center of the circle is (-2,3) and the radius is 3 units.