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For problem 8, the given information is:

- Center at the origin (0, 0)
- A focus at \(\left(-\frac{11}{2}, 0\right)\)
- A vertex at \(\left(-\frac{13}{5}, 0\right)\)

This describes a horizontal ellipse since both the focus and vertex lie along the x-axis. The equation of an ellipse in standard form is:

\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]

Where:
- \((h, k)\) is the center of the ellipse,
- \(a\) is the distance from the center to a vertex,
- \(b\) is the distance from the center to the co-vertices, and
- \(c\) is the distance from the center to a focus.

From the problem:
- The center is at the origin, so \(h = 0\) and \(k = 0\), making the equation simpler as:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Now, we have:
- A focus at \(\left(-\frac{11}{2}, 0\right)\) means \(c = \frac{11}{2} = 5.5\).
- A vertex at \(\left(-\frac{13}{5}, 0\right)\) means \(a = \frac{13}{5} = 2.6\).

The relationship between \(a\), \(b\), and \(c\) for an ellipse is:

\[
c^2 = a^2 - b^2
\]

Substituting the values of \(a\) and \(c\):

\[
(5.5)^2 = (2.6)^2 - b^2
\]
\[
30.25 = 6.76 - b^2
\]
\[
b^2 = 6.76 - 30.25 = -23.49
\]

This result is not correct, implying an error in interpretation of the numbers or problem parameters. Let's recheck the inputs.

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Explore how to derive the equation of an ellipse with a center at the origin, a focus at (-11/2, 0), and a vertex at (-13/5, 0). Learn the key relationships between the focus, vertex, and the equation of an ellipse. Suitable for students in Grades 10-12.

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