Consider the curves $C_1 :y=\frac{8}{27}x^3$ and $C_2 : y = (x + a)^2$. Find the range of 'a' for which there exists two common tangents to the curves $C_1$ and $C_2$ other than x-axis
My approach for $C_1 :y'=\frac{8}{9}x^2$
For $C_2 :y'=2(x+a)$
Equating both we get $\frac{8}{9}x^2=2(x+a)$ but not able to get the answer
The equation $\frac{8}{9}x^2=2(x+a)$ means the slope of the tangent lines are equal at the same $x$ value, but the common tangent line can be tangent to points with different $x$ of the two curves.
So, let us suppose the common tangent line is tangent to $C_1$ at $(x_1,y_1)$ and to $C_2$ at $(x_2,y_2)$, and let $k$ denote the slope, then we have the following equations: $$ \left\{ \begin{array}{l}k=\frac{8}{9}x_1^2=2(x_2+a) \\ y_1-y_2=k(x_1-x_2) \end{array}\right.$$
Substituting $y_1$ and $y_2$ with the expression of $C_1$ and $C_2$, and by some simplification(noticing that $x_1\neq 0$ because it means the tangent line to be $x$-axis), we get the equation $$ a=\frac{2}{9}(x_1^2-3x_1) $$ then by letting $x_1$ range over $\mathbb{R}-\{0\}$, we know that the range of $a$ is $[-\frac{1}{2},+\infty]$.