Question

Write the length of the latus-rectum of the hyperbola 16x² − 9y² = 144.

Answer 1

Equation of the hyperbola:

\[16 x^2 - 9 y^2 = 144\]

This equation can be rewritten in the following way:

\[\frac{16 x^2}{144} - \frac{9 y^2}{144} = 1\]

\[ \Rightarrow \frac{x^2}{9} - \frac{y^2}{16} = 1\]

\[ \Rightarrow \frac{x^2}{3^2} - \frac{y^2}{4^2} = 1\]

This is the standard form of a hyperbola with \[a = 3\] and \[b = 4\].

Length of the latus rectum = \[\frac{2 b^2}{a}\]

Substituting the value of a and b, we get:
Length of the ​latus rectum  \[= \frac{2 \times 4^2}{3} = \frac{32}{3}\]