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


Sagot :

To determine which equation represents a circle with a center at [tex]\((2, -8)\)[/tex] and a radius of 11, we need to use the standard form of the equation of a circle. The standard form is given by:

[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 is [tex]\((2, -8)\)[/tex] and the radius is 11, we can substitute these values into the standard form of the equation.

1. Identify the center and radius:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = -8\)[/tex]
- [tex]\(r = 11\)[/tex]

2. Substitute these values into the standard form:
[tex]\[ (x - 2)^2 + (y + 8)^2 = 11^2 \][/tex]

3. Simplify the right-hand side of the equation:
[tex]\[ (x - 2)^2 + (y + 8)^2 = 121 \][/tex]

Therefore, the equation that represents 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]

Let's verify:

- The given options are:
- [tex]\((x - 8)^2 + (y + 2)^2 = 11\)[/tex]
- [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]
- [tex]\((x + 2)^2 + (y - 8)^2 = 11\)[/tex]
- [tex]\((x + 8)^2 + (y - 2)^2 = 121\)[/tex]

Out of these options, the second option [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex] is the correct one.
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