Let A(4,-4) and B(9,6) be points on the parabola y^2=4x.

Question

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola $y^2=4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $	riangle ACB$ is maximum. Then, the area (in sq. units) of $	riangle ACB$, is:

Accepted Answer

Parametrize $C=(t^2,2t)$ ($A:t=-2,B:t=3$). The area is maximized when the tangent at $C$ is parallel to chord $AB$.

Slope $AB=\dfrac{6+4}{9-4}=2$; tangent slope at $C$ is $\dfrac1t=2\Rightarrow t=\dfrac12$, so $C=\left(\dfrac14,1\right)$.

$$\text{Area}=\frac12\left|4(6-1)+9(1+4)+\frac14(-4-6)\right|=\frac12\left|20+45-2.5\right|=\frac{62.5}2=31.25=31\frac14.$$

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