Question:
The eccentricity of the hyperbola whose latus rectum is $8$ and conjugate axis is equal to half of the distance between the foci is
The eccentricity of the hyperbola whose latus rectum is $8$ and conjugate axis is equal to half of the distance between the foci is
Updated On: Apr 16, 2024
- $ \frac{4}{3}$
- $ \frac{4}{\sqrt{3}}$
- $ \frac{2}{\sqrt{3}}$
- None of these
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The Correct Option is C
Solution and Explanation
Let equation of hyperbola be $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Given, $\frac{2b^{2}}{a}=8$
$\Rightarrow \frac{b^{2}}{a}=4$
and $2b=\frac{1}{2}\left(2ae\right)$
$\Rightarrow 2b = ae$
$\Rightarrow 4b^{2} = a^{2}e^{2}$
$\Rightarrow\, 4\left(\frac{b^{2}}{a^{2}}\right)=e^{2}$
$\Rightarrow 4\left(e^{2}-1\right)=e^{2} \left[\because\, b^{2}=a^{2} \left(e^{2}-1\right)\right]$
$\Rightarrow 3e^{2} = 4$
$\Rightarrow e=\frac{2}{\sqrt{3}}$
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Concepts Used:
Hyperbola
Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.
Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.
