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

The correct option is B $y=\frac{32}{9}x$

Given hyperbola may be written as $\frac{x^2}{9}-\frac{y^2}{16}=1$
$\Rightarrow a^2=9,b^2=16$
Slope of line $x=2y$ is $\frac{1}{2}$
We know If $m$ and $m^{\prime }$ are the slopes of conjugate diameters then, $m{m}^{\prime }=\frac{b^2}{a^2}=\frac{16}{9}$
$\Rightarrow m^{\prime }=\frac{32}{9}$
Therefore required line is $y=m^{\prime }x=\frac{32}{9}x$
Hence. option 'B' is correct.