Question

The area bounded by the curve $y^2=4a^2(x-1)$ and the lines $x=1$ and $y=4a$ is equal to :

• A. $\frac{16a}{3}$ sq unit
• B. $5a$ sq unit
• C. $\frac{17a}{4}$ sq unit
• D. None of these

Answer 1

Answer: (A)

Given curve : $y^2=4a^2(x-1)$
Lines : $x=1$ and $y=4a$
Now, $(y-0{)}^2=4a^2(x-1)$
This is a parabola with vertex $A(1,0)$.

Required area
= area of shaded region ABC
$=\underset{1}{\overset{5}{\int }}$ [y(line) - y(parabola)]dx
$=\underset{1}{\overset{5}{\int }}\left[4a-2a\sqrt{x-1}\right]dx$
$=\underset{1}{\overset{5}{\int }}{\left[4ax-2a\times \frac{2}{3}{\left(x-1\right)}^{32}\right]}_1^5$
$=\left(20a-\frac{32a}{3}\right)-\left(4a-0\right)=\frac{16}{3}a$ sq.unit