Question

What is the area of the parabola $y^2=4bx$ bounded by its latus rectum?

• A. $\frac{2b^2}{3}$ square unit
• B. $\frac{4b^2}{3}$ square unit
• C. $b^2$ square unit
• D. $\frac{8b^2}{3}$ square unit

Answer 1

The correct option is D $\frac{8b^2}{3}$ square unit

For parabola $y^2=4bx$

latus rectum in a line is b

Area required = $2{\int }_0^{b}ydx$

=$2{\int }_0^b\sqrt{4bx}dx$

= $2\sqrt{4b}{\int }_0^b\sqrt{x}dx$

= $4\sqrt{b}\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]$

= $\frac{8}{3}\sqrt{b}{b}^{\frac{3}{2}}$

= $\frac{8}{3}{b}^{\frac{4}{2}}$

area = $\frac{8}{3}{b}^2$ sq.units.