Question

The vertices of a right angled triangle lies on a rectangular hyperbola $xy=4$. The angle between the tangent at the right angled vertex and the hypotenuse of the triangle is $\frac{\alpha \pi }{12}$ , then $\alpha$ is

Answer 1

Let $ABC$ be a right angled triangle
with $\angle ACB={90}^{\circ }$
Let $A=(2t_1,\frac{2}{t_1}),B=(2t_2,\frac{2}{t_2}),C=(2t_3,\frac{2}{t_3})$
$m_{AB}=-\frac{1}{t_1{t}_2},m_{BC}=-\frac{1}{t_2{t}_3},m_{AC}=-\frac{1}{t_1{t}_3}$
$\angle ACB={90}^{\circ }\Rightarrow t_3^2=-\frac{1}{t_1{t}_2}$
Slope of $AB=t_3^2$
Slope of tangent at $C=-\frac{1}{t_3^2}$
$\Rightarrow$ Tangent is perpendicular to the hypotenuse
$\Rightarrow \frac{k\pi }{12}=\frac{\pi }{2}$
$\Rightarrow k=6$