Question

A hyperbola passes through $(3,3)$ and the length of its conjugate axis is $8$. The length of the latus rectum is:

• A. $\frac{20}{3}$
• B. $\frac{40}{3}$
• C. $\frac{50}{3}$
• D. None of these

Answer 1

The correct option is B

$\frac{40}{3}$

Explanation for the correct option:

Step 1. Find the length of the latus rectum:

Given that:

Length of Conjugate Axis, $2b=8$

$\Rightarrow$ $\begin{array}{rcl}b& =& \frac{8}{2}\\ & =& 4\end{array}$

Equation of Hyperbola is: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

If it passes through point $\left(3,3\right)$, so $x=3$ and $y=3$

Step 2. Substituting the values in Equation of hyperbola, we get

$\frac{3^2}{a^2}-\frac{3^2}{4^2}=1$

$\Rightarrow$ $\frac{9}{a^2}-\frac{9}{16}=1$

$\Rightarrow$ $9\left(\frac{1}{a^2}-\frac{1}{16}\right)=1$

$\Rightarrow$$9\left(\frac{16}{a^2}-\frac{a^2}{16}\right)=1$

$\Rightarrow$ $9\left(\frac{16-a^2}{a^{2}16}\right)=1$

$\Rightarrow$ $9\left(16-a^2\right)=16a^2$

$\Rightarrow$ $144-9a^2=16a^2$

$\Rightarrow$ $144=16a^2+9a^2$

$\Rightarrow$ $144=25a^2$

$\Rightarrow$ $a^2=\frac{144}{25}$

$\Rightarrow$ $a=\frac{12}{5}$

As we know, Length of the Latus rectum $\mathbf{=}\mathbf{}\left(\frac{2b^2}{a}\right)$

Step 3. Substituting the values of $a$ and $b$:

$$\begin{gathered} =\left(\frac{2{\left(4\right)}^2}{{\displaystyle \frac{12}{5}}}\right) \\ =\left(\frac{2\times 16\times 5}{12}\right) \\ =\frac{160}{12} \\ =\frac{40}{3} \end{gathered}$$

$\therefore$The Length of Latus Rectum is $\left(\frac{40}{3}\right)$

Hence, the correct Option is (B).