Hyperbola Focus and Directrix Relations
The standard equation of a hyperbola centered at the origin is given by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. For this hyperbola, the foci are located at $(\pm ae, 0)$ and the corresponding directrices are $x = \pm \frac{a}{e}$.
Determining Eccentricity (e)
We are given a focus of the hyperbola H as $(\sqrt{10}, 0)$ and the corresponding directrix as $x = \frac{9}{\sqrt{10}}$. From the focus, we have:
$ae = \sqrt{10} \quad (1)$
From the directrix, we have:
$\frac{a}{e} = \frac{9}{\sqrt{10}} \quad (2)$
To find the eccentricity $e$, we can divide equation (1) by equation (2):
$\frac{ae}{a/e} = \frac{\sqrt{10}}{9/\sqrt{10}}$
$e^2 = \frac{\sqrt{10} \times \sqrt{10}}{9} = \frac{10}{9}$
Thus, $e^2 = \frac{10}{9}$.
Calculating Latus Rectum (l)
First, let's find the value of $a$. Using equation (1) and $e^2 = \frac{10}{9}$ (which means $e = \frac{\sqrt{10}}{3}$):
$a \left( \frac{\sqrt{10}}{3} \right) = \sqrt{10}$
$a = 3$
Next, we find $b^2$ using the relation $b^2 = a^2(e^2 - 1)$:
$b^2 = (3)^2 \left( \frac{10}{9} - 1 \right)$
$b^2 = 9 \left( \frac{10 - 9}{9} \right) = 9 \left( \frac{1}{9} \right) = 1$
The length of the latus rectum, $l$, is given by the formula $l = \frac{2b^2}{a}$:
$l = \frac{2(1)}{3} = \frac{2}{3}$
Final Calculation
We need to calculate the value of $9(e^2 + l)$. We have $e^2 = \frac{10}{9}$ and $l = \frac{2}{3}$.
$9(e^2 + l) = 9 \left( \frac{10}{9} + \frac{2}{3} \right)$
To add the fractions, find a common denominator:
$9 \left( \frac{10}{9} + \frac{2 \times 3}{3 \times 3} \right) = 9 \left( \frac{10}{9} + \frac{6}{9} \right)$
$9 \left( \frac{10 + 6}{9} \right) = 9 \left( \frac{16}{9} \right)$
$9 \times \frac{16}{9} = 16$
Therefore, $9(e^2 + l) = 16$.
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