Question

If the area of a square is cut off by the parabola passing through the two adjacent vertices of the square and touching mid-point of one of its sides is $k$ times the area of square, then the value of $k$ is

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\frac{2}{3}$

Let the parabola $x^2=4ay$ and the sides of square are $x=\pm 8a,y=0$ and $y=16a$
The region cut off by parabola is shown below :
Area of square $=(16a{)}^2$
and Area cut off by parabola
$=2\underset{0}{\overset{16a}{\int }}\sqrt{4ay}dy=2[2\sqrt{a}\cdot \frac{2}{3}(16a{)}^{3/2}]$
$=\frac{2}{3}(16a{)}^2$ $=k\cdot$ area of square
$=k(16a{)}^2\Rightarrow k=\frac{2}{3}$