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 follow these steps:
1. Identify key components:
- The center [tex]\((h, k)\)[/tex] of the circle is [tex]\((4, 0)\)[/tex].
- One of the points on the circle is [tex]\((-2, 8)\)[/tex].
2. Calculate the radius using the distance between the center and the point on the circle:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute [tex]\((x_1, y_1) = (4, 0)\)[/tex] and [tex]\((x_2, y_2) = (-2, 8)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{((-2) - 4)^2 + (8 - 0)^2}
\][/tex]
Simplify inside the square root:
[tex]\[
= \sqrt{((-6)^2 + 8^2)}
= \sqrt{(36 + 64)}
= \sqrt{100}
= 10
\][/tex]
Hence, the radius [tex]\( r \)[/tex] of the circle is 10.
3. Write the standard form equation of the circle using the calculated radius [tex]\( r \)[/tex] and the center [tex]\((h, k)\)[/tex]:
The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substitute [tex]\( h = 4 \)[/tex], [tex]\( k = 0 \)[/tex], and [tex]\( r = 10 \)[/tex]:
[tex]\[
(x - 4)^2 + y^2 = 10^2
\][/tex]
Simplify:
[tex]\[
(x - 4)^2 + y^2 = 100
\][/tex]
4. Match the given equations to the derived equation:
- [tex]\((x - 4)^2 + y^2 = 100\)[/tex]
- [tex]\((x - 4)^2 + y^2 = 10\)[/tex]
- [tex]\(x^2 + (y - 4)^2 = 10\)[/tex]
- [tex]\(x^2 + (y - 4)^2 = 100\)[/tex]
The equation [tex]\((x - 4)^2 + y^2 = 100\)[/tex] matches the derived equation.
Therefore, the equation that represents the circle is:
[tex]\[
(x - 4)^2 + y^2 = 100
\][/tex]
