Question

The point $(2a,a)$ lies inside the region bounded by the parabola $x^2=4y$ and its latus rectum. Then,

• A. $0\le a\le 1$
• B. $0<a<1$
• C. $0<a\le 1$
• D. None of these

Answer 1

The correct option is B $0<a<1$

Let $S\equiv x^2-4y$

Since the point $(2a,a)$ lies inside the parabola,

$\therefore S_{(2a,a)}=4a^2-4a<0\Rightarrow 4a(a-1)<0\Rightarrow a(a-1)<0$ ...(1)

Also, the vertex $A(0,0)$ and the point $(2a,a)$

are on the same side of the line $y=1$ (the equation of the latus rectum)

So, $a-1<0\Rightarrow a<1$ ....(2)

From (1) and (2), we have $a(a-1)<0\Rightarrow 0<a<1$