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 $P{G}^2+Q{g}^2=$

• 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 $-\frac{ax}{\sin \theta }-\frac{by}{\cos \theta }=a^2-b^2....$(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,
$P{G}^2+Q{g}^2={\left\{\frac{\left(a^2-b^2\right)\cos \theta }{a}-a\cos \theta \right\}}^2+$
$(0-b\sin \theta {)}^2+{\left\{\frac{\left(a^2-b^2\right)\cos \theta }{a}-a\cos \theta \right\}}^2+(0-b\cos 0{)}^2$
$=\frac{{\left(a^2-b^2\right)}^2}{a^2}+b^2+a^2$
$=a^2\left\{\frac{{\left(a^2-b^2\right)}^2}{a^4}+\frac{b^2}{a^2}+1\right\}$
$=a^2\left\{{\left(1-\frac{b^2}{a^2}\right)}^2+\frac{b^2}{a^2}+1\right\}=a^2\left(e^4+2-e^2\right)$