Let a line y=mx (m0) intersect the parabola y^2=x at a point P, other than the origin.

Question

Let a line $y=mx$ ($m>0$) intersect the parabola $y^2=x$ at a point $P$, other than the origin. Let the tangent to it at $P$ meet the $x$-axis at the point $Q$. If area$(	riangle OPQ)=4$ sq. units, then $m$ is equal to \_\_\_\_\_\_\_\_.

Answer: 0.5

Accepted Answer

Let $P=(t^2,t)$. Tangent at $P$: $2ty=x+t^2$; at $y=0$: $Q=(-t^2,0)$.

$$\text{Area}(\triangle OPQ)=\frac12|t|\cdot|t|^2=\frac{|t|^3}2=4\Rightarrow|t|=2.$$

$P=(4,2)$, and since $y=mx$ passes through $P$: $m=\dfrac24=\dfrac12=0.5$.

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