If 2x - y + 1 = 0 is a tangent to the hyperbola x ^ 2 a ^ 2 - y ^ 2...

$∴ {\left(1\right)}^{2} = 4{a}^{2} - 16 ⇒ {a}^{2} = \dfrac{17}{4}$

$⇒ a = \dfrac{\sqrt{17}}{2}$

For $2a, \text{4, 1}$, sides are $\sqrt{17}, 4, 1$ ($⇒$ Right angled triangle)

For $\text{2a, 8, 1}$, sides are $\sqrt{17}, 8, 1$ ($⇒$Triangle is not possible)

For $\text{a, 4, 1}$, sides are $\dfrac{\sqrt{17}}{2}, 4, 1$ ($⇒$Triangle is not possible)

For $\text{a, 4, 2}$, sides are $\dfrac{\sqrt{17}}{2}, 4, 2$ ($⇒$Triangle exist but not right angled)