Two straight lines having variable slopes m _ 1 ~and& m _ 2 pass...

Let, the lines are $y = {m}_{1} \left(x - a\right)$ and y = {m}_{2} \left(x + a\right)$<br /><br />which intersects at the point \left(h, k\right) .$

$⇒ k = {m}_{1}\left(h - a\right)$ & k = {m}_{2} \left( h + a\right)$<br /><br />$⇒ k^{2} = {m}_{1}{m}_{2} \left(h^{2} - a^{2}\right)

$⇒$\dfrac{k^{2}}{2} = h^{2} - a^{2}(given,&{m}_{1}{m}_{2} = 2$)<br /><br />$⇒ \dfrac{x^{2}}{1} - \dfrac{y^{2}}{2} = a^{2}$(Hyperbola)<br /><br />Eccentricity$= \sqrt{1 + \dfrac{2}{1}} = \sqrt{3}