QJEEM19Ma0c50: JEE Main 2019 Mathematics - Class 11 JEEM19Ma0c50 Solution: If a directrix of a hyperbola centred at the origin and passing …

Question

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If a directrix of a hyperbola centred at the origin and passing through the point (4, –2

\sqrt 3

) is 5x = 4

\sqrt 5

and its eccentricity is e, then :

Accepted Answer

5x = 4

\sqrt 5

\Rightarrow

x = {4 \over {\sqrt 5 }}

Equation of hyperbola

{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1

it passes through

\left( {4, - 2\sqrt 3 } \right)

\therefore {e^2} = 1 + {{{b^2}} \over {{a^2}}}

\Rightarrow {a^2}{e^2} - {a^2} = {b^2}

\Rightarrow {{16} \over {{a^2}}} - {{12} \over {{a^2}{e^2} - {a^2}}} = 1

\Rightarrow {4 \over {{a^2}}}\left[ {{4 \over 1} - {3 \over {{e^2} - 1}}} \right] = 1

\Rightarrow 4{e^2} - 4 - 3 = ({e^2} - 1)\left( {{{{a^2}} \over 4}} \right)

\Rightarrow 4(4{e^3} - 7) = ({e^2} - 1){\left( {{{4e} \over {\sqrt 5 }}} \right)^2}

\Rightarrow 4{e^4} - 24{e^2} + 35 = 0

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