If the ellipse x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 meets the line (x)/(7)...

Given,

$\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} = 1$ meets the line $\dfrac{x}{7} + \dfrac{y}{2\sqrt{6}} = 1$ on the $x$-axis, so at $x -$axis $y = 0$ putting in both equation and comparing we get, $a = 7$

And $\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} = 1$ meets the line $\dfrac{x}{7} - \dfrac{y}{2\sqrt{6}} = 1$ on the $y$-axis where $x = 0$ so putting in both equation and comparing we get, $b = 2\sqrt{6}$

Therefore, ${e}^{2} = 1 - \dfrac{{b}^{2}}{{a}^{2}} = 1 - \dfrac{24}{49}$

$⇒ e = \sqrt{\dfrac{25}{49}} = \dfrac{5}{7}$