A hyperbola passes through the foci of the ellipse x^225+y^216=1 and its transverse and co…

Question

A hyperbola passes through the foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is

Accepted Answer

$${e_1} = \sqrt {1 - {{16} \over {25}}}  = {3 \over 5}$$ foci ($$ \pm $$ae, 0)

Foci = ($$ \pm $$3, 0)

Let equation of hyperbola be $${{{x^2}} \over {{A^2}}} - {{{y^2}} \over {{B^2}}} = 1$$

Passes through ($$ \pm $$3, 0)

A2 = 9, A = 3, $${e_2} = {5 \over 3}$$

$${e_2}^2 = 1 + {{{B^2}} \over {{A^2}}}$$

$${{25} \over 9} = 1 + {{{B^2}} \over 9} \Rightarrow {B^2} = 16$$

Equation of the hyperbola

$${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$$

Solutions · 1