Question

Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola.

16x² – 9y² = -144

Answer 1

Given: 16x² – 9y² = -144 

To find: eccentricity(e), coordinates of the foci f(m, n), equation of directrix, length of latus-rectum of hyperbola.

Formula used: 

For hyperbola \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = -1\)

Eccentricity(e) is given by,

e = \(\frac{C}{b}\), Where c = \(\sqrt{a^2 +b^2}\)

Foci is given by (0, ± be) 

The equation of directrix are: y = ± \(\frac{b}{e}\)

Length of latus rectum is \(\frac{2a^2}{b}\)

Here, a = 3 and b = 4

Foci: (0, ±5) 

The equation of directrix are:

Length of latus rectum,

= \(\frac{2a^2}{b}\)

= \(\frac{2 \times (3)^2}{4}\)

= \(\frac{9}{2}\)