To determine the center and the radius of the circle given by the equation [tex]\((x+1)^2 + y^2 = 25\)[/tex], we'll compare it to the standard form of the 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
- [tex]\(r\)[/tex] is the radius of the circle
Now, let's match the given equation [tex]\((x+1)^2 + y^2 = 25\)[/tex] with the standard form:
1. Identify the terms for the center:
- The term [tex]\((x + 1)^2\)[/tex] can be written as [tex]\((x - (-1))^2\)[/tex]. So, [tex]\(h = -1\)[/tex].
- The term [tex]\(y^2\)[/tex] can be written as [tex]\((y - 0)^2\)[/tex]. So, [tex]\(k = 0\)[/tex].
Therefore, the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-1, 0)\)[/tex].
2. Identify the radius:
- The right-hand side of the equation is [tex]\(25\)[/tex]. This corresponds to [tex]\(r^2\)[/tex] in the standard form.
- So, [tex]\(r^2 = 25\)[/tex].
- To find [tex]\(r\)[/tex], we take the square root of [tex]\(25\)[/tex]:
[tex]\[
r = \sqrt{25} = 5
\][/tex]
Thus, the center of the circle is [tex]\((-1, 0)\)[/tex] and the radius of the circle is [tex]\(5\)[/tex].
Summarizing:
- Center: [tex]\((-1, 0)\)[/tex]
- Radius: [tex]\(5\)[/tex]
