How many points with Integer x and Y co-ordinate lies within the circl

S1. Circle touches parabola at only 1 point. We do not have any idea about where the centre of the circle is.
Hence, insufficient

S2. We know only the centre. Can't say anything about at how many points does the circle touches or cuts the parabola.
Hence, insufficient

Combining S1 & S2


Circle touches parabola at only 1 point. So, that means they just touch each other. For the parabola, if x=0, y=4. So the parabola intersects the y-axis at (0,4).

This is the point where circle touched the parabola. Even if 'a' is any value > 0.

Circle of circle is given as (0,0). Now, we have got the circle with eqn x^2+y^2=16. (4 is the radius of circle).

For this circle the integer (x,y) inside the circle can be all those point satisfying

x^2+y^2<16

So, all the pairs in the 1st quadrant lying inside the circle will be

(1,1) (1,2) (2,2) (2,1) (3,1) (3,2) (2,3) (1,3) - Total 8.

Similarly in 4 quadrants it will be 8*4=32.

We also have (0,1) (1,0) (0,-1) (-1,0) (0,0) making the count go to 37.

Yes we can say about the number of points lying inside the circle and that number is 37.

Hence, C will be the answer.