Question

The locus of the poles of the normal chords of the rectangular hyperbola $xy=c^2$is:

• A. $(x^2-y^2{)}^2+4c^{2}xy=0$
• B. $(x^2+y^2{)}^2+4c^{2}xy=0$
• C. $(x^2-y^2{)}^2-4c^{2}xy=0$
• D. $(x^2+y^2{)}^2-4c^{2}xy=0$

Answer 1

The correct option is A $(x^2-y^2{)}^2+4c^{2}xy=0$

When $xy=c^2$, the asymptotes are the coordinate axes, they have a parametric representation $x=ct$ and $y=\frac{c}{t}$

Equation of directrix of parabola are $x+y=\pm \sqrt{2}c$

$y-\frac{c}{t}=t^2(x-ct)(x+y)(x-y)={(\pm \sqrt{2}c)}^2\Rightarrow x^2-y^2=\pm 2c^2$

Squaring on both sides

${(x^2-y^2)}^2=-4c^{2}xy\therefore {(x^2-y^2)}^2+4c^{2}xy=0$