Question

Tangent is drawn to a rectangular hyperbola $xy=c^2$.
The area of the triangle formed by tangents
and its intercepts is $72$ sq. units.

• A. $c^2=36$
• B. $xy=36$
• C. Equation of tangent at $(6,6)$ is $x+y=6$
• D. Equation of tangent at $(6,6)$ is $2x-y=6$

Answer 1

Answer: (A), (B), (C)

Equation of tangent at $(6,6)$ is $x+y=6$

We know that
Area of triangle formed by tangent to hyperbola $xy=c^2$ and its intercept is constant and equal to $2c^2.$
So
$\Rightarrow 2c^2=72\Rightarrow c^2=36$

Equation of rectangular hyperbola:
$xy=36$

Equation of tangent on hyperbola
$xy=c^2$ at the point $P(x_1,y_1)$ is
$\Rightarrow x{y}_1+y{x}_1=c^2$
Given, $P(x_1,y_1)\equiv (6,6)$
$\Rightarrow 6x+6y=36$
$\Rightarrow x+y=6$