Question

The area bounded by the parabola y² = 4ax, latusrectum and x-axis is ___________ .

पर्याय

• 0

• \[\frac{4}{3} a^2\]

• \[\frac{2}{3} a^2\]

• \[\frac{a^2}{3}\]

Answer 1

\[\frac{4}{3} a^2\]

 

Clearly, the latusrectum passes x-axis through the point D(a, 0).
Therefore, the required area ABCD,
\[A = \int_0^a y d x ...........\left(\text{Where, } y = 2\sqrt{ax} \right)\]
\[ = \int_0^1 2\sqrt{ax} d x\]
\[ = \left[ \frac{4\sqrt{a}}{3} \left( x \right)^\frac{3}{2} \right]_0^a \]
\[ = \left[ \frac{4\sqrt{a}}{3} \left( a \right)^\frac{3}{2} \right] - \left[ \frac{4\sqrt{a}}{3} \left( 0 \right)^\frac{3}{2} \right]\]
\[ = \frac{4}{3} a^2\text{ square units }\]