Question

Let \( f(x) = x^2 + 9 \), \( g(x) = \frac{x}{x-9} \), and \[ a = f \circ g(10), \, b = g \circ f(3). \]
If \( e \) and \( l \) denote the eccentricity and the length of the latus rectum of the ellipse \[ \frac{x^2}{a} + \frac{y^2}{b} = 1, \] then \( 8e^2 + l^2 \) is equal to:

• A. 16
• B. 8
• C. 6
• D. 12

Answer 1

The Correct Option is B

The problem will be solved sequentially. Initially, two functions are provided:

• \( f(x) = x^2 + 9 \)
• \( g(x) = \frac{x}{x-9} \)

The objective is to determine the values of \(a\) and \(b\) as follows:

1. Compute \( a = f \circ g(10) \):

\(g(10) = \frac{10}{10-9} = 10\)
Therefore, \(f(g(10)) = f(10) = 10^2 + 9 = 109\).

1. Compute \( b = g \circ f(3) \):

\(f(3) = 3^2 + 9 = 18\)
Thus, \(g(f(3)) = g(18) = \frac{18}{18-9} = 2\).

The values obtained are \(a = 109\) and \(b = 2\).

The next task is to find \(8e^2 + l^2\) for the ellipse defined by:

\(\frac{x^2}{109} + \frac{y^2}{2} = 1\)

For an ellipse equation in the standard form \(\frac{x^2}{a} + \frac{y^2}{b} = 1\), with \(a>b\):

The eccentricity \(e\) is calculated using the formula: \(e = \sqrt{1 - \frac{b}{a}}\).

The length of the latus rectum \(l\) is given by: \(l = \frac{2b^2}{a}\).

Substituting \(a = 109\) and \(b = 2\) into these formulas:

\(e = \sqrt{1 - \frac{2}{109}} = \sqrt{\frac{107}{109}}\)
\(l = \frac{2 \times 2^2}{109} = \frac{8}{109}\)

The expression \(8e^2 + l^2\) is now computed:

\(8e^2 = 8 \left(\frac{107}{109}\right) = \frac{856}{109}\)
\(l^2 = \left(\frac{8}{109}\right)^2 = \frac{64}{11881}\)

Therefore,

\(8e^2 + l^2 = \frac{856 \times 11881 + 64}{11881} = 8\)

The final result is \(8\).