Question:

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

Updated On: Jul 7, 2022
  • $y=\frac{16}{9}x$
  • $y=\frac{32}{9}x$
  • $x=\frac{16}{9}y$
  • $x=\frac{32}{9}y$
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The Correct Option is B

Solution and Explanation

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 $y =\frac{32}{9}x$
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Notes on Hyperbola

Concepts Used:

Hyperbola

Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.

Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.

Hyperbola