Question

Equation of common tangent to ellipse $5x^2+2y^2=10,$ and hyperbola $11x^2-3y^2=33$ is

• A. $y=\pm 3x\pm 21$
• B. $y=\pm x\pm 3$
• C. $y=\pm 4x\pm 37$
• D. $3x\pm y=12$

Answer 1

Answer: (C)

$y=\pm 4x\pm 37$

$Formulaused$
$equationof\tan gent{x}^2/a^2-y^2/b^2=1$
$y=mx\pm \sqrt{a^2{m}^2-b^2}$
$equationof\tan gent{x}^2/a^2+y^2/b^2=1$
$y=mx\pm \sqrt{a^2{m}^2+b^2}$
$Given$
$5x^2+2y^2=10$
$\Rightarrow \frac{x^2}{2}+\frac{y^2}{5}=1$
$a^2=2,b^2=5$
$equationof\tan gent$
$y=mx\pm \sqrt{a^2{m}^2+b^2}$
$=mx\pm \sqrt{2m^2+5}.......\left(i\right)$
$11x^2-3y^2=33$
$\Rightarrow \frac{x^2}{3}-\frac{y^2}{11}=1$
$a^2=3,b^2=11$
$equationof\tan gent$
$y=mx\pm \sqrt{a^2{m}^2+b^2}$
$=mx\pm \sqrt{3m^2-11}........\left(ii\right)$
$\mathrm{Sin}ce\tan gentiscommon,henceequating\left(i\right)and\left(ii\right)$
$mx\pm \sqrt{2m^2+5}=mx\pm \sqrt{3m^2-11}$
$\Rightarrow \sqrt{2m^2+5}=\sqrt{3m^2-11}$
$squaringbothsides$
$2m^2+5=3m^2-11$
$\Rightarrow m^2=16$
$m=\pm 4$
$Henceequationof\tan gent$
$y=\pm 4x\pm \sqrt{37}$