Question

The locus of the foot of the perpendicular from the centre of the hyperbola $xy=c^2$ on a variable tangent is :

• A. $(x^2-y^2{)}^2=4c^{2}xy$
• B. $(x^2+y^2{)}^2=2c^{2}xy$
• C. $(x^2-y^2)=4c^{2}xy$
• D. $(x^2+y^2{)}^2=4c^{2}xy$

Answer 1

The correct option is D $(x^2+y^2{)}^2=4c^{2}xy$

Equation of the tangent at $(ct,\frac{c}{t})$ to the hyperbola $xy=c^2$
is $x+t^{2}y=2ct$
If $(h,k)$ is the foot of perpendicular from the center $(0,0)$
on the tangent then $-\frac{1}{t^2}+\frac{k}{h}=-1$
Also $h=t^2,k=2ct$
Eliminating $t$ we get the locus
${(x^2+y^2)}^2={4c}^{2}xy$