Q. The equations of tangents to the ellipse $3x^{2} + 4y^{2} = 5$, which are inclined at $30{^\circ}$ 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
Explanation
(d)
Given equation of ellipse is
$\frac{x^{2}}{\frac{5}{3}} + \frac{y^{2}}{\frac{5}{4}} = 1$
The equation of tangents in slope from is
$y = mx \pm \sqrt{\frac{5}{3}m^{2} + \frac{5}{4}}$
Slope of tangents are $\frac{1}{\sqrt{3}}$ or $- \frac{1}{\sqrt{3}}$
$\therefore y = \pm \frac{1}{\sqrt{3}}x \pm \sqrt{\frac{5}{9} + \frac{5}{4}}$
$\Rightarrow y = \pm \frac{1}{\sqrt{3}}x \pm \frac{\sqrt{65}}{6}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.