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

• A. $4$
• B. $8$
• C. $10$
• D. $12$

Answer 1

The correct option is B $8$

Let $(\lambda ,\mu )$ be interior to the circle, if
${\lambda }^2+{\mu }^2-16<0$
$\Rightarrow \lambda =\{\pm 0,\pm 1,\pm 2,\pm 3\},\mu =\{\pm 0,\pm 1,\pm 2,\pm 3\}$
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
Hence, 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.
$\therefore (1,0),(2,0),(3,0),(1,1),(2,1),(2,1),(2,2),(3,2)$ are the possible points.
Hence total $8$ Points.