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

Explanation:

Step1: Complete the square for x - terms

The coefficient of x is - 8. Half of it is - 4, and its square is 16. Add 16 to both sides of the equation $x^{2}-8x + y^{2}-9 = 0$.
$x^{2}-8x + 16+y^{2}-9=16$

Step2: Rewrite the left - hand side in factored form

Using the perfect - square formula $(a - b)^2=a^{2}-2ab + b^{2}$, where $a = x$ and $b = 4$, we have $(x - 4)^{2}+y^{2}=25$. This is the standard form of the circle equation $(x - h)^{2}+(y - k)^{2}=r^{2}$, where $(h,k)$ is the center of the circle and r is the radius.

Step3: Identify the center and radius

Comparing $(x - 4)^{2}+y^{2}=25$ with $(x - h)^{2}+(y - k)^{2}=r^{2}$, we get $h = 4$, $k = 0$, and $r^{2}=25$, so $r = 5$.

Answer:

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