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  • Tell me? Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation 9e2−18e+5=0. If S(5, 0) is a focus and 5x=9 is the corresponding directrix of this hyperbola, then a2

Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation 9e2−18e+5=0.  If S(5, 0) is a focus and 5x=9 is the corresponding directrix of this hyperbola, then a2−b2 is equal to :

  • Option 1)

    7

  • Option 2)

    -7

  • Option 3)

    5

  • Option 4)

    -5

 

Answers (1)

best_answer

As we learnt in 

Equation of directrices -

x=\pm \frac{a}{e}

- wherein

For the Hyperbola

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

 9e^2-18e+5=0

\Rightarrow 9e^2-15e-3e+5=0

\Rightarrow 3e \left ( 3e-5 \right )-1 \left ( 3e-5 \right )=0

e= \frac{1}{3}  or

e= \frac{5}{3}

for Hyperbola    e= \frac{5}{3}

Also  ae-\frac{a}{e}=5 - \frac{9}{5}=\frac{16}{5}

\frac{5}{3}a-\frac{3}{5}a= \frac{16}{5}

\frac{16a}{15}= \frac{16}{5}

\Rightarrow a=3

Also, b^2= a^2\left ( e^2-1 \right )= 9 \left ( \frac{25}{9} -1 \right )=16

a^2-b^2=-7


Option 1)

7

Incorrect

Option 2)

-7

Correct

Option 3)

5

Incorrect

Option 4)

-5

Incorrect

Posted by

Aadil

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