Question

The area of the segment cut off from the parabola is

$x^2=4y$ by the line x - 2y + 4 = 0

• A. 9
• B. 12
• C. 16
• D. 24

Answer 1

The correct option is A 9

Parabola given $x^2=4y$

Straight line is x - 2y + 4 = 0

$y=\frac{x+4}{2}$

$\Rightarrow x^2=2(x+4)$

$\Rightarrow x^2-2x-8=0$

$\Rightarrow x^2-4x+2x-8=0$

$\Rightarrow x(x-4)+2(x-4)=0$

$\Rightarrow x=4,-2$

Required area = POQ

$=\underset{-2}{\overset{4}{\int }}ydx$ from straight line $-\underset{-2}{\overset{4}{\int }}ydx$ from parabola

$=\underset{-2}{\overset{4}{\int }}\left(\frac{x+4}{2}\right)-\underset{-2}{\overset{4}{\int }}\frac{x^2}{4}dx$

$=\frac{1}{2}{|\frac{x^2}{2}+4x|}_{-2}^4-\frac{1}{4}{\left|\frac{x^3}{3}\right|}_{-2}^4$

$=\frac{1}{2}\{8+16-(-6\left)\right\}-\frac{1}{12}(64+8)$

$=\frac{1}{2}\times 30-\frac{1}{12}\times 72=15-6=9$