<s>[INST] Intersection of two tangents on a parabola proof There are two tangent lines on a parabola $x^2$. The $x$ values of where the tangent lines intersect with the parabola are $a$ and $b$ respectively. The point where the two tangent lines intersect has an $x$ value of $c$. 
Prove that $c=(a+b)/2$
I have tried taking the derivative of $x^2$ and using the point slope form to find each tangent line's equation, and then making each tangent equal to each other to find that $c$ is equal to $a+b$, but each time I try it out I can't get anything out of the algebra
 [/INST] Consider the curve $C$ with the equation $y=x^2$. When $x=a$, we have $y=a^2$. The derivative is given by $y'=2x$ and so the gradient at the is exactly $2a$. The equation of the tangent line is 
$$y-a^2=2a(x-a)$$
Expanding this gives $y-2ax=-a^2$. Similarily, the equation of the tangent line to $C$ at the point with $x$-coordinate $x=b$ is given by $y-2bx=-b^2$. We are left needing to solve
\begin{eqnarray*}
2ax-y&=&a^2 \\ 2bx-y&=&b^2 
\end{eqnarray*}
Subtracting the second equation from the first gives $2(a-b)x=a^2-b^2$. Notice that $a^2-b^2$ factorises to give $(a-b)(a+b)$ and hence: $2(a-b)x=(a-b)(a+b)$. Assuming $a \neq b$ we can divide both sides by $a-b$ to give $2x=a+b$ or, better still $x=\tfrac{1}{2}(a+b).$
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