If the area (in sq.

Question

If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x$, $\lambda>0$, is $\dfrac19$, then $\lambda$ is equal to:

Accepted Answer

Setting $y=\lambda x$ in $y^2=4\lambda x$: $x=0,\dfrac4\lambda$.

$$\text{Area} = \int_0^{4/\lambda}\left(2\sqrt{\lambda x}-\lambda x\right)dx = \left[\frac{4\sqrt\lambda}{3}x^{3/2}-\frac{\lambda x^2}{2}\right]_0^{4/\lambda} = \frac{32}{3\lambda}-\frac{8}{\lambda} = \frac{8}{3\lambda}$$

$$\frac{8}{3\lambda} = \frac19 \implies \lambda = 24$$

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