Q1 Let $a > 0$. Consider two curves

$$C_1: y = e^{6x}$$
$$C_2: y = ax^2.$$

We are to find the condition on $a$ such that there exist two straight lines, each of which is tangent to both $C_1$ and $C_2$.

The equation of the tangent to $C_1$ at a point $(t, e^{6t})$ is

$$y = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$

This is tangent also to $C_2$ under the condition that the quadratic equation

$$ax^2 = \mathbf{A}e^{6t}x - e^{6t}(\mathbf{B}t - \mathbf{C})$$

has just one solution. Hence, the equation

$$\mathbf{D}e^{12t} - ae^{6t}(\mathbf{E}t - \mathbf{F}) = 0$$

must hold for $a$ and $t$. From this equation we obtain

$$a = \frac{\mathbf{D}}{\mathbf{E}t - \mathbf{F}}e^{6t}$$

Let $f(t)$ denote the right side of this equation. The condition under which there exist two straight lines each of which is tangent to both $C_1$ and $C_2$, is that the straight line $s = a$ intersects the graph of $s = f(t)$ at two points.

Now, the derivative of $f(t)$ is

$$f'(t) = \frac{108e^{6t}(\mathbf{G}t - \mathbf{H})}{(\mathbf{E}t - \mathbf{F})^2}.$$

Hence the condition on $a$ that we are seeking is

$$a > \square e^{\square}.$$

Note that $\lim_{t \to \infty} \frac{e^t}{t} = \infty$.