Let the eccentricity of the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$\sqrt{\frac{5}{2}}$$ and length of its latus rectum be $$6\sqrt{2}$$. If $$y = 2x + c$$ is a tangent to the hyperbola $$H$$, then the value of $$c^2$$ is equal to

We are given a hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ with eccentricity $$e = \sqrt{\frac{5}{2}}$$ and latus rectum $$= 6\sqrt{2}$$.

For a hyperbola, $$e^2 = 1 + \frac{b^2}{a^2}$$. Substituting $$\frac{5}{2} = 1 + \frac{b^2}{a^2}$$ gives $$\frac{b^2}{a^2} = \frac{3}{2}$$, so $$b^2 = \frac{3a^2}{2}$$. The length of the latus rectum is $$\frac{2b^2}{a}$$; setting $$\frac{2b^2}{a} = 6\sqrt{2}$$ yields $$\frac{2 \cdot \frac{3a^2}{2}}{a} = 6\sqrt{2}$$ and hence $$3a = 6\sqrt{2}$$, so $$a = 2\sqrt{2}$$. Therefore $$a^2 = 8$$ and $$b^2 = \frac{3 \times 8}{2} = 12$$.

For a line $$y = mx + c$$ to be tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$, the condition is $$c^2 = a^2 m^2 - b^2$$. Substituting $$m = 2$$, $$a^2 = 8$$ and $$b^2 = 12$$ gives $$c^2 = 8 \times 4 - 12 = 32 - 12 = 20$$. Therefore, the correct answer is Option B: $$20$$.