Let L be a common tangent line to the curves 4 x ^ 2 + 9 y ^ 2 = 36...

Given curves are $\dfrac{{x}^{2}}{9} + \dfrac{{y}^{2}}{4} = 1$

${x}^{2} + {y}^{2} = \dfrac{31}{4}$

let slope of common tangent be $m$

so tangents are $y = mx \pm \sqrt{9{m}^{2} + 4}$

$y = mx \pm \dfrac{\sqrt{31}}{2}\sqrt{1 + {m}^{2}}$

hence $9{m}^{2} + 4 = \dfrac{31}{4}\left(1 + {m}^{2}\right)$

$⇒ 36{m}^{2} + 16 = 31 + 31{m}^{2} ⇒ {m}^{2} = 3$