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Write an equation of a circle that is centered at [tex]\((-3, 2)\)[/tex] with a radius of 5.

A. [tex]\((x-3)^2 + (y+2)^2 = 5\)[/tex]

B. [tex]\((x+3)^2 + (y-2)^2 = 5\)[/tex]

C. [tex]\((x-3)^2 + (y+2)^2 = 25\)[/tex]

D. [tex]\((x+3)^2 + (y-2)^2 = 25\)[/tex]

Sagot :

GinnyAnsweridn
To determine the equation of a circle given the center and the radius, we use the standard form of the equation of a circle:

[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 is [tex]\((-3, 2)\)[/tex]
- The radius of the circle is [tex]\(5\)[/tex]

First, let's substitute the given center [tex]\((-3, 2)\)[/tex] into the general equation. Here, [tex]\(h = -3\)[/tex] and [tex]\(k = 2\)[/tex]:

[tex]\[ (x - (-3))^2 + (y - 2)^2 = r^2 \][/tex]

Simplify the expression inside the parentheses:

[tex]\[ (x + 3)^2 + (y - 2)^2 = r^2 \][/tex]

Next, we substitute the given radius [tex]\(r = 5\)[/tex] into the equation:

[tex]\[ (x + 3)^2 + (y - 2)^2 = 5^2 \][/tex]

Since [tex]\(5^2 = 25\)[/tex], the equation becomes:

[tex]\[ (x + 3)^2 + (y - 2)^2 = 25 \][/tex]

Thus, the correct equation of the circle is:

[tex]\[ (x + 3)^2 + (y - 2)^2 = 25 \][/tex]

Therefore, the correct answer is [tex]\(D\)[/tex]:
[tex]\[ (x + 3)^2 + (y - 2)^2 = 25 \][/tex]
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