Question

The number of integral values of $a$ for which the point $P(a^2,a)$ lies in the region corresponding to the acute angle between the lines $2y=x$ and $4y=x$ is

Answer 1

Given point $P(a^2,a)$ lies in between $x-2y=0$ and $x-4y=0$

Hence $a^2-2a>0$

$a>0$ and $a>2$

On combining both conditions

$a>2---\left(1\right)$

Now $a^2-4a<0$

$a(a-4)<0$

$a<4---\left(2\right)$

On combining (1) and (2)

$2<a<4$

Hence the no of values of a is only $1$