Question

The locus of point of intersection of tangents to the circle $x=a\cos \theta ,y=a\sin \theta$ at points whose parametric angles differ by $\pi /4$ is?

• A. $x^2+y^2=2(\sqrt{2}-1{)}^2{a}^2$
• B. $x^2+y^2=2(2-\sqrt{2})a^2$
• C. $x^2+y^2=(\sqrt{2}+1{)}^2{a}^2$
• D. $9(x^2+y^2)=4a^2$

Answer 1

Answer: (B)

$x^2+y^2=2(2-\sqrt{2})a^2$

The equation of the locus of the point of intersection of the tangent whose parameters

differ by angle x is

$\begin{array}{c}{x}^2+y^2=a^2{\sec }^2\left(\frac{x}{2}\right)\\ x^2+y^2=a^2{\sec }^2\frac{45}{2}\\ =\frac{a^{2}2\sqrt{2}}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}\\ =\frac{a^{2}2\sqrt{2}\times \sqrt{2}-1}{1}\\ \to a^2\left(4-2\sqrt{2}\right)\\ x^2+y^2\to 2a^2\left(2-\sqrt{2}\right)\end{array}$