Question

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

• A. $x^2=\lambda c^2$
• B. $y^2=\lambda c^2$
• C. $xy=\lambda c^2$
• D. $x+y=\lambda c^2$

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 $=t^2$
Let $P(h,k)$ be the point through which the normal is passing.
Then, $c{t}^4-h{t}^3+tk-c=0$
Here $\sum t_i=\frac{h}{c};\sum t_1{t}_2=0$
Hence, sum of the slopes of the normal
$=\sum t_i^2={(\sum t_i)}^2-2\sum t_1{t}_2\Rightarrow \frac{h^2}{c^2}-0=\lambda$
Therefore, required locus is $x^2=\lambda c^2$