Question

The number of points with integral coordinates that lie in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x$, is

• A. $8$
• B. $10$
• C. $16$
• D. None of these

Answer 1

The correct option is A $8$

Let $(\alpha ,\beta )$ be the point with integral coordinates and lying in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x$.
Then, ${\alpha }^2+{\beta }^2-16<0$
and ${\beta }^2-4\alpha <0$
It is clear from the figure that $0<\alpha <4$
$\Rightarrow \alpha =1,2,3$ $[\because \alpha ϵZ]$
When, $\alpha =1$
${\beta }^2<4\alpha$
$\Rightarrow {\beta }^2<4$
$\Rightarrow \beta =0,1$
So, the points are $(1,0)$ and $(1,1)$.
When $\alpha =2$
${\beta }^2<4\alpha$
$\Rightarrow {\beta }^2<8$
$\Rightarrow \beta =0,1,2$
So, the points are $(2,0),(2,1)$ and $(2,2)$.
When $\alpha =3$
${\beta }^2<4\alpha$
$\Rightarrow {\beta }^2<12$
$\Rightarrow \beta =0,1,2,3$
So, the points are $(3,0),(3,1),(3,2)$ and $(3,3)$
Out of these points, $(3,3)$ does not satisfy ${\alpha }^2+{\beta }^2-16<0$.

Thus, the points lying in the region are $(1,0),(1,1),(2,0),(2,1),(2,2),(3,0)(3,1)$ and $(3,2)$.