Question

A tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches at the point P on it in the first quadrant and meets the co-ordinate axes in A and B respectively. If P divides AB in the ration $3:1$ find the equation of the
tangent.

Answer 1

Equation of tangent,

$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$

$A(\frac{a}{\cos \theta },0)$ $B(0,\frac{b}{\sin \theta })$

$P(\frac{a}{4\cos \theta },\frac{3b}{4\sin \theta })=P(a\cos \theta ,b\sin \theta )$

$\frac{a}{4\cos \theta }=a\cos \theta$

${\cos }^2\theta =\frac{1}{4}$

$\Rightarrow \cos \theta =\frac{1}{2}$

$\sin \theta =\frac{\sqrt{3}}{2}$

Equation of tangent ,

$\frac{x}{2a}+\frac{y\sqrt{3}}{2b}=1$

$bx+\sqrt{3}ay=2ab$