Question

The eccentricity of the hyperbola whose latus rectum is half of its transverse axis is
A) $\dfrac{1}{{\sqrt 2 }}$
B) $\sqrt {\dfrac{2}{3}}$
C) $\sqrt {\dfrac{3}{2}}$
D) None of these

Answer 1

Here, we will assume that the equation of hyperbola, $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$. Then we will take the transverse axis is $2a$ and the latus rectum is $\dfrac{{2{b^2}}}{a}$. Then we will simplify the obtained equation using the given conditions and use the fact that eccentricity of a hyperbola is always greater than 1 and can be calculated using the formula, $e = \sqrt {1 + \dfrac{{{b^2}}}{{{a^2}}}}$.

Complete step by step solution:
We are given that hyperbola whose latus rectum is half of its transverse axis.
Let us assume that the equation of hyperbola, $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$.
We know that the general equation of a hyperbola is $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, where $a$ is line segment for the $x$–axis and $b$ is the line segment for the $y$–axis.
Here, the transverse axis is $2a$ and the latus rectum is $\dfrac{{2{b^2}}}{a}$.
According to problems, we have

$$\Rightarrow \dfrac{{2{b^2}}}{a} = \dfrac{1}{2}\left( {2a} \right) \\\ \Rightarrow \dfrac{{2{b^2}}}{a} = a \\\$$

Cross-multiplying the above equation, we get
$\Rightarrow 2{b^2} = {a^2}$
We also know that the eccentricity of a hyperbola is always greater than 1 and can be calculated using the formula, $e = \sqrt {1 + \dfrac{{{b^2}}}{{{a^2}}}}$.
Using the above formula of eccentricity in the above equation, we get
$\Rightarrow 2{a^2}\left( {{e^2} - 1} \right) = {a^2}$
Dividing the above equation on both sides by ${a^2}$, we get

$$\Rightarrow \dfrac{{2{a^2}\left( {{e^2} - 1} \right)}}{{{a^2}}} = \dfrac{{{a^2}}}{{{a^2}}} \\\ \Rightarrow 2\left( {{e^2} - 1} \right) = 1 \\\ \Rightarrow 2{e^2} - 2 = 1 \\\$$

Adding the above equation with 2 on both sides, we get

$$\Rightarrow 2{e^2} - 2 + 2 = 1 + 2 \\\ \Rightarrow 2{e^2} = 3 \\\$$

Dividing the above equation by 2 on both sides, we get

$$\Rightarrow \dfrac{{2{e^2}}}{2} = \dfrac{3}{2} \\\ \Rightarrow {e^2} = \dfrac{3}{2} \\\$$

Taking square root on both sides in the above equation, we get
$\Rightarrow e = \sqrt {\dfrac{3}{2}}$

Therefore, the value is $\sqrt {\dfrac{3}{2}}$.
Hence, option C is the correct answer.

Note:
In solving these types of questions, students find the value from one equation and substitute it in the other equation to find the equation of the hyperbola. The possibility of mistake is not being able to apply the formula and properties of quadratic equations to solve. The key step to solve this problem is by knowing the properties to the general equation of hyperbola $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$, the solution will be very simple.