Question

complete the square and write the given equation in standard form. then give the center and radius of the circle and graph the equation.

$x^{2}-8x + y^{2}-9 = 0$

(simplify your answer.)

the center of the circle is $(4,0)$.
(type an ordered pair.)

the radius of the circle is $r = 5$.

use the graphing tool to graph the circle.

click to enlarge graph

Explanation:

Step1: Rearrange the equation

Rearrange the given equation \(x^{2}-8x + y^{2}-9 = 0\) as \((x^{2}-8x)+y^{2}=9\).

Step2: Complete the square for the \(x\) - terms

For the quadratic expression \(x^{2}-8x\), using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a=x\) and \(2ab = 8x\) (so \(b = 4\)). Then \(x^{2}-8x=(x - 4)^{2}-16\).
Substitute \(x^{2}-8x=(x - 4)^{2}-16\) into \((x^{2}-8x)+y^{2}=9\), we get \((x - 4)^{2}-16+y^{2}=9\).

Step3: Write in standard form

Add \(16\) to both sides of the equation \((x - 4)^{2}-16+y^{2}=9\).
The standard form of the circle equation is \((x - 4)^{2}+y^{2}=25\).

Answer:

The standard form of the circle equation is \((x - 4)^{2}+y^{2}=25\). The center of the circle is \((4,0)\) and the radius \(r = 5\).