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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)

best_answer

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

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