Which equation represents a circle with a center at [tex](2, -8)[/tex] and a radius of 11?

A. [tex](x-8)^2 + (y+2)^2 = 11[/tex]
B. [tex](x-2)^2 + (y+8)^2 = 121[/tex]
C. [tex](x+2)^2 + (y-8)^2 = 11[/tex]
D. [tex](x+8)^2 + (y-2)^2 = 121[/tex]



Answer :

To determine the correct equation for a circle with its center at [tex]\((2, -8)\)[/tex] and a radius of 11, let's use the standard form of the equation for a circle. The formula for the equation of a circle in standard form is:

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.

Given:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((2, -8)\)[/tex],
- The radius [tex]\(r\)[/tex] is 11.

First, substitute [tex]\(h = 2\)[/tex], [tex]\(k = -8\)[/tex], and [tex]\(r = 11\)[/tex] into the standard form equation:

[tex]\[ (x - 2)^2 + (y + 8)^2 = 11^2 \][/tex]

Next, calculate [tex]\(11^2\)[/tex]:

[tex]\[ 11^2 = 121 \][/tex]

So, the equation of the circle becomes:

[tex]\[ (x - 2)^2 + (y + 8)^2 = 121 \][/tex]

Let's now check which of the given options matches this equation:

1. [tex]\((x - 8)^2 + (y + 2)^2 = 11\)[/tex]
2. [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]
3. [tex]\((x + 2)^2 + (y - 8)^2 = 11\)[/tex]
4. [tex]\((x + 8)^2 + (y - 2)^2 = 121\)[/tex]

Upon review, we see that the second option matches our derived equation:

[tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]

Therefore, the correct equation representing a circle with a center at [tex]\((2, -8)\)[/tex] and a radius of 11 is:

[tex]\[ (x - 2)^2 + (y + 8)^2 = 121 \][/tex]