Let's solve this step-by-step.
We are given a point [tex]\((-2, 8)\)[/tex] and a center [tex]\((4, 0)\)[/tex], and we want to find the equation of the circle that passes through the given point and has the specified center.
### Step 1: Calculate the radius of the circle
To find the radius, we use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, [tex]\((x_2, y_2) = (-2, 8)\)[/tex] and [tex]\((x_1, y_1) = (4, 0)\)[/tex].
Substitute these values into the distance formula:
[tex]\[
\text{Radius} = \sqrt{(-2 - 4)^2 + (8 - 0)^2} = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
So, the radius [tex]\( r \)[/tex] of the circle is 10.
### Step 2: Write the equation of the circle
The general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In this case, the center [tex]\((h, k)\)[/tex] is [tex]\((4, 0)\)[/tex] and the radius [tex]\( r \)[/tex] is 10. Plugging these values into the general equation:
[tex]\[
(x - 4)^2 + (y - 0)^2 = 10^2
\][/tex]
Simplified, this becomes:
[tex]\[
(x - 4)^2 + y^2 = 100
\][/tex]
### Step 3: Match the equation with given options
Upon comparison with the provided options:
- [tex]\((x - 4)^2 + y^2 = 100\)[/tex]
We find that this equation matches our derived equation.
Therefore, the correct equation representing the circle that contains the point [tex]\((-2, 8)\)[/tex] and has a center at [tex]\((4, 0)\)[/tex] is:
[tex]\[
(x - 4)^2 + y^2 = 100
\][/tex]
