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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

Samar Imam Zaidi
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1 Answers1

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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 $[-\frac{1}{2},+\infty]$.

Simon
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  • I'd look closer at $a=-\frac12$ where two tangents coincide and $a=0$ where there's higher contact between the curves and with the $x$-axis. BTW, the dual curves are $x^2+4axy+4y=0$ and $y+x^3/2=0$ with grobner basis containing $y(32a^3y^2+(-24a-16)y-1)$, from which the discriminant of the quadratic factor is $64(a+2)^2(2a+1)$. – Jan-Magnus Økland Oct 22 '19 at 09:06