Question

If tan ${\theta }_1\tan {\theta }_2=-\frac{a^2}{b^2}$ then the chord joining two points ${\theta }_1$ and ${\theta }_2$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ will subtend a right angle at

• A. focus
• B. centre
• C. end of the major axis
• D. end of the minor axis

Answer 1

Answer: (B)

Given $tan{\theta }_{1}tan{\theta }_2=-\frac{a^2}{b^2},$
Let $P\left({\theta }_1\right),Q\left({\theta }_2\right)$ be two points on the ellipse.
Slope of $CP,CQ$ are $\frac{bsin{\theta }_1}{acos{\theta }_1};\frac{bsin{\theta }_2}{acos{\theta }_2}$
i.e., $\frac{b}{a}tan{\theta }_1,\frac{b}{a}tan{\theta }_2$
Product of the slopes $=\frac{b^2}{a^2}tan{\theta }_{1}tan{\theta }_2$
$=\frac{b^2}{a^2}\left(-\frac{a^2}{b^2}\right)=-1$
$\therefore$ chord will subtend a right angle at the centre.