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If the center of a circle is at [tex]\((5, -3)\)[/tex] and its radius is 4, complete its equation:

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

Sagot :

GinnyAnsweridn
To find the equation of a circle with a given center and radius, we use the standard form of the equation of a circle, which is:

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

Here, [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] represents the radius.

Given:
- The center of the circle is [tex]\((5, -3)\)[/tex]; therefore, [tex]\(h = 5\)[/tex] and [tex]\(k = -3\)[/tex].
- The radius of the circle is [tex]\(4\)[/tex]; thus, [tex]\(r = 4\)[/tex].

Now, let's substitute these values into the standard form of the equation.

1. Substitute [tex]\(h = 5\)[/tex]:
[tex]\[ (x - 5)^2 \][/tex]

2. Substitute [tex]\(k = -3\)[/tex]:
[tex]\[ (y - (-3))^2 = (y + 3)^2 \][/tex]

3. Calculate [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = 4^2 = 16 \][/tex]

Therefore, the complete equation of the circle is:

[tex]\[ (x - 5)^2 + (y + 3)^2 = 16 \][/tex]
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