Question

Circles are drawn on chords of the rectangular hyperbola $xy=c^2$ parallel to the line $y=x$ as diameters. All such circles pass through two fixed points whose co-ordinates are

• A. $(-c,-c)$
• B. $(c,c)$
• C. $(-c,c)$
• D. $(c,-c)$

Answer 1

Answer: (C), (D)

$(c,-c)$

Let the extrimities of the chord be $(c{t}_1,\frac{c}{t_1})$ and $(c{t}_2,\frac{c}{t_2})$ , then
$\frac{\frac{c}{t_2}-\frac{c}{t_1}}{c{t}_2-c{t}_1}=1\Rightarrow t_1{t}_2=-1t_1=\frac{-1}{t_2}=t$
The circle will be
$(x-ct)(x+\frac{c}{t})+(y-\frac{c}{t})(y+ct)=0$
$\Rightarrow (x^2+y^2–2c^2)+c(1/t–t)(x+y)=0$
which is a family of circles
Fixed point will be the intersection of
$x^2+y^2=x^3$ ... (i)
and $x+y=0$ ... (ii)
The fixed points will be $(c,–c)$ and $(–c,c)$