Question

If any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes intercepts $p$ and $q$ on the coordinate axes, then $\frac{a^2}{p^2}+\frac{b^2}{q^2}=$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$1$

The equation of tangent to ellipse is $bxcos\theta +aysin\theta =ab$

The $x$ intercept is $x=\frac{a}{cos\theta }$

The $y$ intercept is $q=\frac{b}{sin\theta }$

$\Rightarrow \frac{a^2}{p^2}+\frac{b^2}{q^2}=\frac{a^2}{(\frac{a}{cos\theta }{)}^2}+\frac{b^2}{{\left(\frac{b}{sin\theta }\right)}^2}={\cos }^2\theta +{\sin }^2\theta =1$

Therefore correct option is $A$