To solve the given equation of an ellipse and find the necessary parameters, let's analyze the equation step-by-step:
[tex]\[
\frac{(x+4)^2}{25}+\frac{(y+2)^2}{9}=1
\][/tex]
This is the standard form of an ellipse equation given by:
[tex]\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\][/tex]
In this equation:
- [tex]\((h, k)\)[/tex] is the center of the ellipse.
- [tex]\(a\)[/tex] is the length of the semi-major axis.
- [tex]\(b\)[/tex] is the length of the semi-minor axis.
See how the given equation matches this standard form.
1. Identify the center of the ellipse [tex]\((h, k)\)[/tex]:
- In the term [tex]\((x + 4)^2\)[/tex], comparing with [tex]\((x - h)^2\)[/tex], we see that [tex]\(h = -4\)[/tex].
- In the term [tex]\((y + 2)^2\)[/tex], comparing with [tex]\((y - k)^2\)[/tex], we see that [tex]\(k = -2\)[/tex].
Therefore, the center [tex]\((h, k)\)[/tex] of the ellipse is:
[tex]\[
(-4, -2)
\][/tex]
2. Find the lengths of the semi-major and semi-minor axes:
- The term [tex]\(\frac{(x + 4)^2}{25}\)[/tex] tells us that [tex]\(a^2 = 25\)[/tex]. Thus, [tex]\(a = \sqrt{25} = 5\)[/tex], which is the length of the semi-major axis.
- The term [tex]\(\frac{(y + 2)^2}{9}\)[/tex] tells us that [tex]\(b^2 = 9\)[/tex]. Thus, [tex]\(b = \sqrt{9} = 3\)[/tex], which is the length of the semi-minor axis.
Summarizing, the parameters for the ellipse described by the given equation are:
- Center: [tex]\((-4, -2)\)[/tex]
- Semi-major axis length: [tex]\(5\)[/tex]
- Semi-minor axis length: [tex]\(3\)[/tex]
