Question

The locus of the point of intersection of the straight lines $\frac{x}{a}+\frac{y}{b}=\lambda$ and $\frac{x}{a}-\frac{y}{b}=\frac{1}{\lambda }$ ($\lambda$ is variable) is?

Answer 1

Assume that the required point of intersection is $(h,k)$.

$\therefore \frac{h}{a}+\frac{k}{b}=\lambda$ and $\frac{h}{a}-\frac{k}{b}=\frac{1}{\lambda }$

To obtain locus, we need to eliminate any parameters. Hence, on multiplying both equations, we have

$(\frac{h}{a}+\frac{k}{b})(\frac{h}{a}-\frac{k}{b})=\lambda \times \frac{1}{\lambda }$

$\therefore \frac{h^2}{a^2}+\frac{hk}{ab}-\frac{hk}{ab}-\frac{k^2}{b^2}=1$

$\therefore \frac{h^2}{a^2}-\frac{k^2}{b^2}=1$

Hence, the required locus is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.