Question

The number of points with non-negative 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

Answer 1

Let $(\lambda ,\mu )$ be interior to the circle.
$\Rightarrow {\lambda }^2+{\mu }^2-16<0$
$\Rightarrow \lambda \in \{0,1,2,3\},\mu \in \{0,1,2,3\}$ as the coordinates are non-negative.

And for $(\lambda ,\mu )$ to be interior to the parabola,${\mu }^2-4\lambda <0$
$\Rightarrow \lambda >{\left(\frac{\mu }{2}\right)}^2$
From circle condition,
if $\mu =0,\lambda =1,2,3$
if $\mu =1,\lambda =1,2,3$
if $\mu =2,\lambda =2,3$
if $\mu =3,\lambda$ has no integral values.
$\therefore (1,0),(2,0),(3,0),(1,1),(2,1),(3,1),(2,2),(3,2)$ are the possible points.
Hence, total $8$ points.