Question

Two straight lines having variable slopes $m_{1}$ and $ \, m_{2}$ pass through the fixed points $\left(a , 0\right)$ and $\left(- a , 0\right)$ respectively. If $m_{1}m_{2}=2$ , then the eccentricity of the locus of the point of intersection of the lines is

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $2$
• D. $\sqrt{\frac{3}{2}}$

Answer 1

Answer: (B)

Let, the lines are $y=m_{1}\left(x - a\right)$ and $y=m_{2}\left(x + a\right)$
which intersects at the point $\left(h , k\right).$
$\Rightarrow k=m_{1}\left(h - a\right)$ & $ \, k=m_{2}\left(h + a\right)$
$\Rightarrow k^{2}=m_{1}m_{2}\left(h^{2} - a^{2}\right)$
$\Rightarrow \frac{k^{2}}{2}=h^{2}-a^{2}$ (given, $m_{1}m_{2}=2$ )
$\Rightarrow \frac{x^{2}}{1}-\frac{y^{2}}{2}=a^{2}$ (Hyperbola)
Eccentricity $=\sqrt{1 + \frac{2}{1}}=\sqrt{3}$