A hyperbola passes through the point P\left ( \sqrt{2} ,\sqrt{3}\right )and has foci at (±2, 0).

  Then the tangent to this hyperbola at P also passes through the point :

  • Option 1)

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

  • Option 2)

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

  • Option 3)

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

  • Option 4)

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

 

Answers (2)

As we learnt in 

Equation of Tangent to Hyperbola -

\frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}= 1

- wherein

For the Hyperbola

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

P\left ( x_{1} ,y_{1}\right )

 

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

Focus is (\pm 2,0) \: ae=2;

b^2 = a^2(e^2-1)\Rightarrow a^2+b^2=4

Hyperbola passes through (\sqrt{2},\sqrt{3})

\frac{2}{a^2}-\frac{3}{b^2} =1; hence \: a^2=1, b^2=3

Equation of hyperbola is \frac{x^2}{1}-\frac{y^2}{3} =1

Equation of tangent is \sqrt{2}x -\frac{\sqrt{3}y}{3}=1

Hence (2\sqrt{2}, 3\sqrt{3}) satisfied it.


Option 1)

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

Correct

Option 2)

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

Incorrect

Option 3)

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

Incorrect

Option 4)

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

Incorrect

N neha

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