Question

Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .

 4x² − 3y² = 36

Answer 1

 Equation of the hyperbola:  4x² − 3y² = 36 

This can be rewritten in the following way:

\[\frac{4 x^2}{36} - \frac{3 y^2}{36} = 1\]

\[\frac{x^2}{9} - \frac{y^2}{12} = 1\]

This is the standard equation of a hyperbola, where  \[a^2 = 9 \text { and } b^2 = 12\] .

\[\Rightarrow b^2 = a^2 ( e^2 - 1)\]

\[ \Rightarrow 12 = 9( e^2 - 1)\]

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

\[ \Rightarrow e^2 = \frac{7}{3}\]

\[ \Rightarrow e = \sqrt{\frac{7}{3}}\]

Coordinates of the foci are given by  \[\left( \pm ae, 0 \right)\]  i.e.

\[\left( \pm \sqrt{21}, 0 \right)\] .

Equation of the directrices: \[x = \pm \frac{a}{e}\]

\[\Rightarrow x = \pm \frac{3}{\sqrt{\frac{7}{3}}}\]

\[ \Rightarrow \sqrt{7}x \pm 3\sqrt{3} = 0\]

Length of the latus rectum of the hyperbola is  \[\frac{2 b^2}{a}\] .

\[\Rightarrow \frac{2 \times 12}{3} = 8\]