Question

A line through the origin meets the circle $x^2+y^2=a^2$ at P and the hyperbola $x^2–y^2=a^2$ at Q. The locus of the point of intersection of the tangent at P to the circle and with the tangent at Q to the hyperbola is

• A.
• B.
• C.
• D. None

Answer 1

The correct option is B

The line through origin is y = mx, it meets the circle $x^2-y^2=a^2$ at P$(\frac{a}{\sqrt{1+m^2}},\frac{am}{\sqrt{1+m^2}})$
Equation of tangent at P is S = 0 $\Rightarrow x+my=a\sqrt{1+m^2}\to \left(1\right)$
y = mx meets the hyperbola $x^2-y^2=a^2$ at Q$(\frac{a}{\sqrt{1+m^2}},\frac{am}{\sqrt{1+m^2}})$
Equation of tangent at Q is x – my = a$\sqrt{1+m^2}\to \left(2\right)$
$(1{)}^2-(2{)}^2\Rightarrow m=\frac{2xy}{a^2}$
$\therefore \left(1\right)\Rightarrow x+\frac{2xy}{a^2}y=a\sqrt{1+\frac{4x^2{y}^2}{a^2}}$$\Rightarrow (a^2+2y^2)x=a$$\sqrt{a^4+4x^2{y}^2}$$\Rightarrow$ $(a^4+4y^4)x^2=a^6$