Question

The equations of tangents to the ellipse $3x^2+4y^2=5$, which are inclined at ${30}^o$ to the x-axis are :

• A. $y=\sqrt{3}x\pm \frac{5}{2}$
• B. $y=\frac{1}{\sqrt{3}}x\pm \frac{5}{2}$
• C. $y=\frac{1}{\sqrt{3}}x\pm 1$
• D. None of these

Answer 1

The correct option is D None of these

Given equation of ellipse is

$\frac{x^2}{5/3}+\frac{y^2}{5/4}=1$

The equation of tangents in slope form is :

$y=mx\pm \sqrt{a^2{m}^2+b^2}$

Here, $a^2=\frac{5}{3}$, $b^2=\frac{5}{4}$

$y=mx\pm \sqrt{\frac{5}{3}{m}^2+\frac{5}{4}}$

$m=\tan {30}^0=\pm \frac{1}{\sqrt{3}}$

Slope of tangents are $\frac{1}{\sqrt{3}}$ or $-\frac{1}{\sqrt{3}}$.

Therefore,

$y=\pm \frac{1}{\sqrt{3}}x\pm \sqrt{\frac{5}{9}+\frac{5}{4}}$

$y=\pm \frac{1}{\sqrt{3}}x\pm \frac{\sqrt{65}}{6}$