Question

The tangent at the point $P$ on the rectangular hyperbola $xy=k^2$ with $C$ intersects the coordinate axes at $Q$ and $R$ . Locus of the circumcentre of triangle $CQR$ is

• A. $x^2+y^2=2k^2$
• B. $x^2+y^2=k^2$
• C. $xy=k^2$
• D. none of these

Answer 1

The correct option is C $xy=k^2$

Given:A tangent at the point $P$ on the rectangular hyperbola $xy=k^2$ with $C$ intersects the coordinate axes at $Q$ and $R$ where $C(0,0)$ is the center of hyperbola.

$\therefore △CQR$ is a rightangled triangle where $\angle C={90}^{\circ }$

The circumcentre of a right angled triangle is the mid-point of its hypotenuse.

Coordinates of hyperbola $xy=k^2$ is $(kt,\frac{k}{t})$

The tangent at $(kt,\frac{k}{t})$ is $x+y{t}^2=2kt$

cuts the $y-$ axis at $Q$

At $x=0\Rightarrow y{t}^2=2kt$

$\Rightarrow y=\frac{2kt}{t^2}=\frac{2k}{t}$

$\therefore$co-ordinates of $Q$ are $(0,\frac{2k}{t})$

At $x-$axis we have $y=0$ at $R$

$\therefore x+0=2kt$

$\Rightarrow x=2kt$

Hence coordinates of $R$ is $(2kt,0)$

Midpoint of $QR$ is $(\frac{0+2kt}{2},\frac{\frac{2k}{t}+0}{2})=(kt,\frac{k}{t})$

$\therefore x=kt,y=\frac{k}{t}$

$\Rightarrow xy=kt\times \frac{k}{t}=k^2$

Hence the locus of the circumcentre of the triangle $CQR$ is $xy=k^2$