An ellipse passes through the foci of the hyperbola,9x^{2}-4y^{2}= 36

and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively.  If the product of eccentricities of the two conics is \frac{1}{2} then which of the following points does not lie on the ellipse ?

  • Option 1)

    \left ( \sqrt{13},0 \right )

  • Option 2)

    \left ( \frac{\sqrt{39}}{2}, \sqrt{3}\right )

  • Option 3)

    \left ( \frac{1}{2}\sqrt{13}, \frac{\sqrt{3}}{2}\right )

  • Option 4)

    \sqrt{\frac{13}{2}},\sqrt{6}

 

Answers (1)

As we learnt in 

Eccentricity -

e= \sqrt{1-\frac{b^{2}}{a^{2}}}

- wherein

For the ellipse  

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

 

 

Coordinates of foci -

\pm ae,o

- wherein

For the ellipse  

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

 

 9x^{2}-4y^{2}=36

=> \frac{x^{2}}{4}-\frac{y^{2}}{9}=1

e=\sqrt{\frac{1+9}{4}}=\frac{\sqrt{13}}{2}

e_{1}\times e _{2}=\frac{1}{2}

\Rightarrow e_{2}=\frac{1}{\sqrt{13}}

Also ae_{1}=2\times \frac{\sqrt{13}}{2}=\sqrt{13}

Hence ellipse passes through \left (\pm\sqrt{13, 0} \right )

 For\: ellipse\:\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1

A^{2}=13

Also, B^{2} =A^{2}\left ( 1-e_{2}^{2} \right )

B^{2} = 13\left ( 1-\frac{1}{13} \right )=12

\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1

=> \frac{x^{2}}{13}+\frac{y^{2}}{12}=1

Here\left ( \frac{1\sqrt{13}}{2},\frac{\sqrt{3}}{2} \right )

doesn't lie on this ellipse


Option 1)

\left ( \sqrt{13},0 \right )

Incorrect option    

Option 2)

\left ( \frac{\sqrt{39}}{2}, \sqrt{3}\right )

Incorrect option    

Option 3)

\left ( \frac{1}{2}\sqrt{13}, \frac{\sqrt{3}}{2}\right )

Correct option

Option 4)

\sqrt{\frac{13}{2}},\sqrt{6}

Incorrect option    

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