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A common tangent to the conics $x^{2}=6y$ and $2x^{2}-4y^{2}=9$ is :
Option: 1
$x-y=\frac{3}{2}$

Option: 2
$x+y=1$

Option: 3
$x+y=\frac{9}{2}$

Option: 4
$x-y=1$

Answers (1)

best_answer

the equation of the parabola is x² = 6y.

or $x^2=4\times \left ( \frac{3}{2} \right )y$

The equation of the tangent is $x=\frac{y}{m}+am$

where m is slope and a = 3/2.

Equation of tangent becomes $2mx = 2y+3m^2$

$y=mx-\frac{3}{2}m^2$

$\\\text { For } 2 x^{2}-4 y^{2}=9 \\ \frac{x^{2}}{\frac{9}{2}}-\frac{y^{2}}{\frac{9}{4}}=1 \\ \\\text{equation of tangent is, }y=m x+\sqrt{\left(a^{2} m^{2}-b^{2}\right)}$

${y}={m x}+\sqrt{\frac{{9 m}^{2}}{2}-\frac{9}{4}}$

Bothe the equation represents the same line

$\\\frac{9m^2}{2}-\frac{9}{4}=\frac{9}{4}m^4\\m^4-2m^2+1=0\\m^2=1$

$x-y=\frac{3}{2}$

Posted by

SANGALDEEP SINGH

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