Question

If the tangent and the normal to a rectangular hyperbola at a point cut off intercept $a_1,a_2$ on one axis and $b_1,b_2$ the other axis then $a_1{a}_2+b_1{b}_2$ is equal to:

• A. $-1$
• B. $0$
• C. $1$
• D. $\sqrt{2}$

Answer 1

The correct option is B $0$

Let the hyperbola be $xy=c^2$
Tangent at any point $(ct,\frac{c}{t})$ is
$x+y{t}^2-2ct=0\Rightarrow \frac{x}{2ct}+\frac{y}{\frac{2c}{t}}=1$
$\therefore a_1=2ct$ and $b_1=\frac{2c}{t}$
Equation of normal is $x{t}^3-yt-c{t}^4+c=0$
$\therefore a_2=\frac{c(t^4-1)}{t^3}$ and $b_2=\frac{-c(t^4-1)}{t}$
$\therefore a_1{a}_2+b_1{b}_2=0$