Question

If rectangular hyperbola $x^2-y^2=4$ is converted to $xy=-c^2$, then the equation of tangent to the hyperbola $xy=-c^2$ at point $t,t$ being a parameter, is :

• A. $\frac{x}{t}+yt=\pm \sqrt{2}$
• B. $\frac{x}{t}+yt\pm 4\sqrt{2}=0$
• C. $\frac{x}{t}-yt\pm 2\sqrt{2}=0$
• D. $\frac{x}{t}-yt\pm 4\sqrt{2}=0$

Answer 1

The correct option is C $\frac{x}{t}-yt\pm 2\sqrt{2}=0$

$x^2-y^2=a^2$ is converted into $xy=-\frac{a^2}{2}$, if axis is rotated by ${45}^{\circ }$ in clockwise direction.

So in given equation,
$c^2=\frac{a^2}{2}=\frac{4}{2}=2$
$c=\pm \sqrt{2}$
Any parametric point on hyperbola $=(ct,\frac{-c}{t})$
Equation of tangent $xct-y\frac{c}{t}=-2c^2$
$\Rightarrow xt-\frac{y}{t}=-2c$

$\Rightarrow xt-\frac{y}{t}\pm 2\sqrt{2}=0$