Problem Solving

How many points with Integer x and Y co-ordinates lie within the circle with centre at origin if the circle intersects with parabola y = ax^2 + 4 where a>0 at only one Point

A) 16
B) 17
C) 36
D) 37
E) 41

Source: https://www.GMATinsight.com

Answer: Option D

This is really more of a Calc I HW problem than a GMAT question, in my opinion.

The vertex of the given parabola is at the point (0,4) and it opens up. In order to circle intersects the parabola at only one point we should have the circle with center at the origin and with radius equal to 4. This circle is given by the equation: x^2+y^2=16.

We now are interested in finding all the integer solutions of the inequality: x^2+y^2<16.

The solutions in the first quadrant are: (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1) and (3,2). (TOTAL=8)

The solutions in the second quadrant are: (-1,1), (-1,2), (-1,3), (-2,1), (-2,2), (-2,3), (-3,1) and (-3,2). (TOTAL=8)

The solutions in the third quadrant are: (-1,-1), (-1,-2), (-1,-3), (-2,-1), (-2,-2), (-2,-3), (-3,-1) and (-3,-2). (TOTAL=8)

The solutions in the fourth quadrant are: (1,-1), (1,-2), (1,-3), (2,-1), (2,-2), (2,-3), (3,-1) and (3,-2). (TOTAL=8)

The solutions in the axes X and Y are: (0,0), (1,0), (2,0), (3,0), (-1,0), (-2,0), (-3,0), (0,1), (0,2), (0,3), (0,-1), (0,-2) and (0,-3). (TOTAL=13)

The number of points with integer x and y co-ordinates is 45. This options is not in the given options.