To determine which equation represents a circle that contains the point [tex]\((-2,8)\)[/tex] and has a center at [tex]\((4,0)\)[/tex], we first need to calculate the radius of the circle.
The distance formula,
[tex]\[
\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2},
\][/tex]
can be used to find the radius of the circle.
1. Identify the coordinates:
- Center of the circle, [tex]\((x_1, y_1) = (4, 0)\)[/tex]
- Given point on the circle, [tex]\((x_2, y_2) = (-2, 8)\)[/tex]
2. Substitute the values into the distance formula:
[tex]\[
\text{radius} = \sqrt{((-2)-4)^2 + (8-0)^2}
\][/tex]
3. Simplify the expression inside the square root:
[tex]\[
(-2-4)^2 = (-6)^2 = 36
\][/tex]
[tex]\[
(8-0)^2 = 8^2 = 64
\][/tex]
4. Sum these results and take the square root:
[tex]\[
\text{radius} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
So, the radius of the circle is 10. The equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In this problem:
- The center [tex]\((h, k) = (4, 0)\)[/tex]
- The radius [tex]\(r = 10\)[/tex]
Plugging these values into the equation of the circle:
[tex]\[
(x - 4)^2 + (y - 0)^2 = 10^2
\][/tex]
Simplifying,
[tex]\[
(x - 4)^2 + y^2 = 100
\][/tex]
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]
