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 $=\sqrt{1+\frac{2}{1}}=\sqrt{3}$