Question

If the tangent and normal to a rectangular hyporbola $xy=c^2$ at a point cuts off intercepts $a_1$ and $a_2$ on $x-$axis and $b_1$, and $b_2$ on the $y-$axis, then $a_1{a}_2+b_1{b}_2=$

• A. $c^2$
• B. $0$
• C. $2c^2$
• D. $\frac{2}{c^2}$

Answer 1

The correct option is B $0$

Let point $P(x_1,y_1)$ on the hyperbola $xy=c^2$
equations of tangents and normals at $P$ on the hyperbola will be
$x{y}_1+y{x}_1=2c^2$ and $x{x}_1-y{y}_1=x_1^2-y_1^2$ respectively
$\Rightarrow a_1=\frac{2c^2}{y_1},b_1=\frac{2c^2}{x_1},a_2=\frac{x_1^2-y_1^2}{x_1},b_2=-\frac{(x_1^2-y_1^2)}{y_1}\therefore a_1{a}_2+b_1{b}_2=0$