Question

The tangent and normal to the ellipse $4x^2+9x^2=36$ at a point P on it meets the major axis in Q and R respectively. If QR = 4, then the eccentric angle of P is

• A. ${\cos }^{-1}\frac{3}{5}$
• B. ${\cos }^{-1}\frac{2}{3}$
• C. ${\cos }^{-1}\frac{1}{3}$
• D. ${\cos }^{-1}\frac{1}{5}$

Answer 1

The correct option is A ${\cos }^{-1}\frac{3}{5}$

$P=(3\cos \theta ,2\sin \theta )$
Equation of tangent at P is $\frac{x\cos \theta }{3}+\frac{y\sin \theta }{2}=1\to \left(1\right)$
Equation of the normal at P is $\frac{3x}{\cos \theta }-\frac{2y}{\sin \theta }=5\to \left(2\right)$
Points of intersection of (1) & (2) with x-axis are $Q(\frac{3}{\cos \theta },0)andR=(\frac{5\cos \theta }{3},0)$
Given that $QR=4\Rightarrow |\frac{3}{\cos \theta }-\frac{5\cos \theta }{3}|=4\Rightarrow \cos \theta =\frac{3}{5}\Rightarrow \theta ={\cos }^{-1}\left(\frac{3}{5}\right)$