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How many points with Integer x and Y co-ordinates lie within the circl

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How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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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

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[Reveal] Spoiler: OA

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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.
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Re: How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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New post 04 Jun 2017, 14:17
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.
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Re: How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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New post 04 Jun 2017, 23:23
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.
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Re: How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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New post 05 Jun 2017, 08:14
Yes, it should be 45 and not 37
11,12,13,21,22,23,31,32-similar points in other coordinates 8*4
01,02,03-similar points in other coordinaters 3*4
00
sum 45
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How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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New post 30 Jun 2017, 06:43
Bunuel Need your help on this one! Do we get questions on parabolas such as above ? I personally haven't encountered . Please provide your solution either ways.
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Re: How many points with Integer x and Y co-ordinates lie within the circl [#permalink]

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New post 13 Jan 2018, 00:23
Can anyone please help with the question above.I, too, agree with the sum of 45.
Re: How many points with Integer x and Y co-ordinates lie within the circl   [#permalink] 13 Jan 2018, 00:23
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How many points with Integer x and Y co-ordinates lie within the circl

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