If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1$, then which of the following cannot be sides of a right angled triangle?

If 2x - y + 1 = 0 is a tangent to the hyperbola x²/a² - y²/16 = 1, then which of the following cannot be sides of a right angled triangle? $\text{Line } 2x - y + 1 = 0 \text{ is tangent to } \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{16} = 1$ $\text{Tangent slope } m = 2, \quad \text{intercept } c = 1, \quad b^{2} = 16$ $\text{Find } a, \text{ then test each set: } \text{(largest)}^{2} \stackrel{?}{=} \text{sum of squares}$ $y = mx + c \text{ tangent to } \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \iff c^2 = a^2m^2 - b^2$ $2x - y + 1 = 0 \Rightarrow y = 2x + 1 \Rightarrow m = 2,\ c = 1$ $1 = 4a^{2} - 16 \Rightarrow a^{2} = \dfrac{17}{4}$ $a^{2} = \dfrac{17}{4}, \qquad a = \dfrac{\sqrt{17}}{2}$ $\text{(1) } a,\ 4,\ 1:\ \text{largest } 4:\ 16 \stackrel{?}{=} \dfrac{17}{4} + 1 = \dfrac{21}{4}$ $\text{(1) }: 16 \neq \dfrac{21}{4} \Rightarrow \text{not right}$ $\text{(2) } a,\ 4,\ 2:\ 16 \neq \dfrac{17}{4} + 4 = \dfrac{33}{4} \Rightarrow \text{not right}$ $\text{(3) } 2a,\ 8,\ 1:\ 64 \neq 4a^{2} + 1 = 17 + 1 = 18 \Rightarrow \text{not right}$ $\text{(4) } 2a,\ 4,\ 1:\ (2a)^{2} = 4a^{2} = 17 = 16 + 1 = 4^{2} + 1^{2}$ $\Rightarrow \text{right triangle (hypotenuse } 2a)$ $\boxed{\text{Correct option: } (4)} \quad (2a)^2 = 4^2 + 1^2$