Question

which equation represents a circle that contains the point $(-2, 8)$ and has a center at $(4, 0)?$
distance formula: $sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$\bigcirc$ $(x - 4)^2 + y^2 = 100$
$\bigcirc$ $(x - 4)^2 + y^2 = 10$
$\bigcirc$ $x^2 + (y - 4)^2 = 10$
$\bigcirc$ $x^2 + (y - 4)^2 = 100$

Explanation:

Step1: Calculate radius via distance formula

Substitute $(x_1,y_1)=(4,0)$ and $(x_2,y_2)=(-2,8)$ into the distance formula:
$$r = \sqrt{(-2 - 4)^2 + (8 - 0)^2}$$

Step2: Simplify to find $r^2$

First compute inside the square root:
$$(-2-4)^2 + (8-0)^2 = (-6)^2 + 8^2 = 36 + 64 = 100$$
Since the circle equation uses $r^2$, we get $r^2=100$.

Step3: Write circle standard equation

Standard form: $(x-h)^2+(y-k)^2=r^2$, where $(h,k)=(4,0)$:
$$(x-4)^2 + (y-0)^2 = 100$$
Simplify to: $(x-4)^2 + y^2 = 100$

Answer:

A. $(x - 4)^2 + y^2 = 100$