Given : A circle, 2 x ^ 2 + 2 y ^ 2 = 5 and a parabola, y ^ 2 = 4 5...

Tangent to ${y}^{2} = 4\sqrt{5}x$ will be $y = mx + \dfrac{\sqrt{5}}{m}$
As tangent to ${x}^{2} + {y}^{2} = \dfrac{5}{2}$ will be $y = mx \pm \dfrac{\sqrt{5}}{\sqrt{2}}\sqrt{1 + {m}^{2}}$
Hence for common tangent

$\dfrac{\sqrt{5}}{m} = \dfrac{\sqrt{5}}{\sqrt{2}}\sqrt{1 + {m}^{2}}$
$⇒ {m}^{4} + {m}^{2} - 2 = 0$

$⇒ {m}^{2} = 1$ , ${m}^{2} = - 2$ (Not Possible)

$⇒ m = \pm 1$
Hence common tangent is $y = x + \sqrt{5}$

Both the statements are correct

The values of $m$ also satisfy the equation in statement - $II$ but it is not a correct explanation for statement - $I$.