mdacosta wrote:
I agree with the 32 figure, but what about points (2,0), (3,0) and (-2,0), (-3,0), etc. I would think it would make the # of points inside the circle 32 + 12 (on the axis's) + 1 (origin) = 45 total points?
14101992 wrote:
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
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) making the count go to 36.
We have an option 36, but wait. What about (0,0). That is also inside the circle.
Hence, the answer will be D =37.
Hi
Yes, I agree with mdacosta .. If you are looking at all the integer points lying within
x^2 + y^2 < 16
Then those are 45, NOT 37.
One way to do is to rewrite the inequality like this:
y^2 < 16 - x^2 Here if x=0, then we have y^2 < 16 Or -4 < y < 4. y can take 7 integer values here (-3, -2, -1, 0, 1, 2, 3)
If x=1, then we have y^2 < 15. Here also y can take 7 integer values (-3, -2, -1, 0, 1, 2, 3)
If x=-1, then also we have y^2 < 15. Here also y can take 7 integer values (-3, -2, -1, 0, 1, 2, 3)
If x=2, then we have y^2 < 12. Here also y can take 7 integer values (-3, -2, -1, 0, 1, 2, 3)
If x=-2, then also we have y^2 < 15. Here also y can take 7 integer values (-3, -2, -1, 0, 1, 2, 3)
If x=3, then we have y^2 < 7. Here y can take 5 integer values (-2, -1, 0, 1, 2)
If x=-3, then also we have y^2 < 7. Here also y can take 5 integer values (-2, -1, 0, 1, 2)
We cannot take x as any other integer value as that would mean y^2 < 0
So total required points with integer coordinates = 7 + 7 + 7 + 7 + 7 + 5 + 5 = 45.