Question

The diameter of $16x^2 - 9y^2 = 144$ which is conjugate to $x = 2y $ is

• A. $y=\frac{16}{9}x$
• B. $y=\frac{32}{9}x$
• C. $x=\frac{16}{9}y$
• D. $x=\frac{32}{9}y$

Answer 1

Answer: (B)

We know that $y = m_1 x, y = m_2x$ are conjugate diameters to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1$ if $m_{1}m_{2} = \frac{b^{2}}{a^{2}}$
Here hyperbola is $\frac{x^{2}}{9}-\frac{y^{2}}{16} = 1$
$\therefore a^{2} = 9, b^{2}=16 \,$and $\, m_{1} = \frac{1}{2} $
$ \therefore \frac{1}{2}\left(m_{2} \right)= \frac{16}{9}$
$ \Rightarrow m_{2} = \frac{32}{9} $
$\therefore $ reqd. diameter is $y =\frac{32}{9}x$