To determine the radius of the circle given its equation [tex]\((x + 5)^2 + (y - 3)^2 = 4^2\)[/tex], we can follow these steps:
1. Identify the Standard Form of a Circle's Equation: The standard form of the equation of a circle 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.
2. Compare with the Given Equation: The given equation is [tex]\((x + 5)^2 + (y - 3)^2 = 4^2\)[/tex]. To compare it with the standard form:
- Notice that [tex]\((x - (-5))^2\)[/tex] can be rewritten as [tex]\((x + 5)^2\)[/tex], so [tex]\(h = -5\)[/tex].
- Also, [tex]\((y - 3)^2\)[/tex] matches the standard form exactly, so [tex]\(k = 3\)[/tex].
3. Determine the Radius [tex]\(r\)[/tex]: The right side of the given equation is [tex]\(4^2\)[/tex], which is in the form [tex]\(r^2\)[/tex]:
- Thus, [tex]\(r^2 = 4^2\)[/tex].
- To find [tex]\(r\)[/tex], take the square root of both sides: [tex]\(r = \sqrt{4^2}\)[/tex].
- Simplifying, [tex]\(r = 4\)[/tex].
Therefore, the radius of the circle is 4 units.
