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 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

Answer: (A)

$x^2=\lambda c^2$

Equation of normal at any point $(ct,\frac{c}{t})$ is ${ct}^4-{xt}^3+ty-c=0$

$\Rightarrow$ Slope of normal $=t^2$

Let $P(h,k)$ be the point through which the normal is passing.

Then,${ct}^4-{ht}^2+tk-c=0$

$\Rightarrow \sum t_i=\frac{h}{c}$ and $\sum t_i{t}_j=0$

Hence,sum of the slopes of the normal $\Rightarrow \sum t_i^2={(\sum t_i)}^2=h^2=c^2\lambda$

Therefore, required locus is $x^2=\lambda c^2$