Question

Find the angle between the tangents drawn from $x^2+y^2=41$ to an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\frac{5\pi }{6}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is B $\frac{\pi }{2}$

We know, locus of point of intersection of tangents, which are at right angles is given by , $x^2+y^2=a^2+b^2$
$\text{Here,}{a}^2=25,b^2=16$

Thus, equation of director circle is given by,
$\Rightarrow x^2+y^2=a^2+b^2$
$\Rightarrow x^2+y^2=25+16$
$\Rightarrow x^2+y^2=41$
This is the same equation to the given equation in question.
Hence, $x^2+y^2=41$ is a director circle

We know, Director circle is the locus of points of intersection of tangents which meet at right angles

Hence, the angle between the tangents drawn from $x^2+y^2=41$ to an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $\frac{\pi }{2}$