To find the radius of the circle whose equation is given by [tex]\((x+5)^2 + (y-3)^2 = 4^2\)[/tex], let's examine the standard form of a circle's equation.
The standard form of a circle's equation 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.
In the given equation [tex]\((x+5)^2 + (y-3)^2 = 4^2\)[/tex], we can compare it with the standard form:
- The term [tex]\((x+5)^2\)[/tex] corresponds to [tex]\((x-h)^2\)[/tex], which means [tex]\(h = -5\)[/tex].
- The term [tex]\((y-3)^2\)[/tex] corresponds to [tex]\((y-k)^2\)[/tex], which means [tex]\(k = 3\)[/tex].
- The right side of the equation [tex]\(4^2\)[/tex] is in the form [tex]\(r^2\)[/tex].
From this comparison, we can identify that [tex]\(r^2 = 4^2\)[/tex].
So, the value of [tex]\(r\)[/tex] (the radius) can be found by taking the square root of [tex]\(4^2\)[/tex]:
[tex]\[
r = \sqrt{4^2}
\][/tex]
Simplifying this, we get:
[tex]\[
r = \sqrt{16}
\][/tex]
Therefore:
[tex]\[
r = 4
\][/tex]
Thus, the radius of the circle is [tex]\(4\)[/tex] units.
To summarize, the correct answer is:
4 units
