Question

The graph of the equation x^2+6x+y^2-16y=-9 is a circle. Choose True or False for each statement

A. The center of the circle is (3,-8). (true or false?)

B. The circle is tangent to the x-axis. (true or false?)

C. The circle has a radius of 64. (true or false?)

Answer 1

see the explanation

Explanation:

we have

`x^2+6x+y^2-16y=-9`

Convert the equation of the circle in center radius form

Group terms that contain the same variable

`(x^2+6x)+(y^2-16y)=-9`

Complete the square twice. Remember to balance the equation by adding the same constants to each side

`(x^2+6x+9)+(y^2-16y+64)=-9+9+64`

`(x^2+6x+9)+(y^2-16y+64)=64`

Rewrite as perfect squares

`(x+3)^2+(y-8)^2=8^2`

The center of the circle is (-3,8)

The radius of the circle is 8 units

Verify each statement

A. The center of the circle is (3,-8).

False

The center of the circle is (-3,8)

B. The circle is tangent to the x-axis

True

The circle is tangent to x=5 and x=-11 and is tangent to y=0 and y=16

Remember that y=0 is the x-axis

C. The circle has a radius of 64

False

The radius of the circle is 8 units