The eccentricity of a hyperbola passing through the points (3,0),… - Vidyadip

MCQ

The eccentricity of a hyperbola passing through the points $(3,0),(3 \sqrt{2}, 2)$ will be

A
$\sqrt{13}$
✓
$\frac{\sqrt{13}}{3}$
C
$\frac{\sqrt{13}}{4}$
D
$\frac{\sqrt{13}}{2}$

✓

Answer

Correct option: B.

$\frac{\sqrt{13}}{3}$

(B)
Let the equation of hyperbola be $\frac{x^2}{ a ^2}-\frac{y^2}{b^2}=1$
Since hyperbola passes through the points $(3,0),(3 \sqrt{2}, 2)$
$\therefore \quad \frac{9}{a^2}-0=1$ and $\frac{18}{a^2}-\frac{4}{b^2}=1$
$\Rightarrow a^2=9$ and $\frac{4}{b^2}=\frac{18}{a^2}-1$
$\Rightarrow a^2=9$ and $\frac{4}{b^2}=\frac{18}{9}-1$
$\Rightarrow a^2=9$ and $b^2=4$
Now, $e =\sqrt{1+\frac{ b ^2}{ a ^2}}=\sqrt{1+\frac{4}{9}}=\frac{\sqrt{13}}{3}$

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