Question

If the normals at $P (\theta)$ and $Q (\pi / 2+\theta)$ to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meet the major axis at $G$ and $g$, respectively, then $PG ^{2}+ Qg ^{2}=$

• A. $b^{2}\left(1-e^{2}\right)\left(2-e^{2}\right)$
• B. $a^{2}\left(e^{4}-e^{2}+2\right)$
• C. $a ^{2}\left(1+ e ^{2}\right)\left(2+ e ^{2}\right)$
• D. $b^{2}\left(1+e^{2}\right)\left(2+e^{2}\right)$

Answer 1

Answer: (B)

Normal at $P (\theta)$ is
$\frac{ ax }{\cos \theta}-\frac{ by }{\sin \theta}= a ^{2}- b ^{2}...$(i)
Normal at $P \left(\frac{\pi}{2}+\theta\right)$ is
$\frac{ ax }{\cos \{(\pi / 2)+\theta\}}-\frac{ ay }{\sin \{(\pi / 2)+\theta\}}= a ^{2}- b$
or $-\frac{a x}{\sin \theta}-\frac{b y}{\cos \theta}=a^{2}-b^{2}....$(ii)
Equation (i) and (ii) meet the major axis at
$G \left(\frac{\left( a ^{2}- b ^{2}\right) \cos \theta}{ a }, 0\right)$
and $g \left(\frac{\left( a ^{2}- b ^{2}\right) \sin \theta}{ a }, 0\right)$
Now,
$ PG ^{2}+ Qg ^{2}=\left\{\frac{\left( a ^{2}- b ^{2}\right) \cos \theta}{ a }- a \cos \theta\right\}^{2}+ $
$(0- b \sin \theta)^{2}+\left\{\frac{\left( a ^{2}- b ^{2}\right) \cos \theta}{ a }- a \cos \theta\right\}^{2}+ (0- b \cos 0)^{2}$
$=\frac{\left(a^{2}-b^{2}\right)^{2}}{a^{2}}+b^{2}+a^{2}$
$=a^{2}\left\{\frac{\left(a^{2}-b^{2}\right)^{2}}{a^{4}}+\frac{b^{2}}{a^{2}}+1\right\}$
$=a^{2}\left\{\left(1-\frac{b^{2}}{a^{2}}\right)^{2}+\frac{b^{2}}{a^{2}}+1\right\}=a^{2}\left(e^{4}+2-e^{2}\right)$