Question

The diameter of \(16x^2- 9 y^2 = 144\) which is conjugate to x = 2y, is

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

Answer 1

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


Diameters $y=m_{1}xandy=m_{2}x$ are conjugate diameters of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, if $m_1{m}_2=\frac{b^2}{a^2}$ .
Here, $a^2=9,b^2=16and{m}_1=\frac{1}{2}$
$\therefore$ $m_1{m}_2=\frac{b^2}{a^2}\Rightarrow \frac{1}{2}\left(m_2\right)=\frac{16}{9}$
$\therefore$ $m_2=\frac{32}{9}$
Thus, the required diameter is $y=\frac{32x}{9}$