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Q)

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

$\begin{array}{1 1}(A)\;y=\large\frac{16 x}{9} \\(B)\;y= \large\frac{32 x}{9} \\(C)\;x= \large\frac{16 y}{9} \\(D)\;x= \large\frac{32 y}{9} \end{array}$

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A)
Diameters $y=m_1x$
$y=m_2x$ are conjugate diameters of the hyperbola $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=1$.
if $m_1m_2= \large\frac{b^2}{a^2}$
Here $a^2=9$
$b^2=16$
$m_1= \large\frac{1}{2}$
$\therefore m_1m_2=\large\frac{b^2}{a^2}$
$\qquad= \large\frac{1}{2}$$ (m_2)$
$\qquad= \large\frac{16}{9}$
$m_2=\large\frac{32}{9}$
Thus the required diameter is $y= \large\frac{32x }{9}$
Hence B is the correct answer.
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