Question

If the sum of the slopes of the normal from a point $P$ to the hyperbola $xy=c^2$ is equal to $\lambda (\lambda \in R^{+})$, then the locus of point $P$ is

• A. $x^2=\lambda c^2$
• B. $y^2=\lambda c^2$
• C. $xy=\lambda c^2$
• D. None of these

Answer 1

The correct option is A $x^2=\lambda c^2$

Equation of normal at any point $(ct,\frac{c}{t})$ is
$c{t}^4-x{t}^3+ty-c=0$
$\Rightarrow$ Slope of normal $=t^2$
Given $\sum t_i^2=\lambda$
Let $P$ be $(h,k)$
$\Rightarrow c{t}^4-h{t}^3+tk-c=0$
$\Rightarrow \sum t_i=\frac{h}{c}$ and $\sum t_i{t}_j=0$
$\Rightarrow \sum t_i^2=(\sum t_i{)}^2$
$\Rightarrow h^2=c^2\lambda$
Therefore, the required locus is $x^2=\lambda c^2$.