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How many points with Integer x and Y coordinates lie within the circl
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Updated on: 13 Apr 2018, 06:53
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How many points with Integer x and Y coordinates 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) 42 D) 45 E) 49 Source: http://www.GMATinsight.com
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Originally posted by GMATinsight on 13 May 2017, 02:37.
Last edited by GMATinsight on 13 Apr 2018, 06:53, edited 1 time in total.




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Re: How many points with Integer x and Y coordinates lie within the circl
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13 May 2017, 06:33
How many points with Integer x and Y coordinates 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 yaxis 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<16So, 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 coordinates lie within the circl
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04 Jun 2017, 15:17
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 coordinates 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 yaxis 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 coordinates lie within the circl
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05 Jun 2017, 00:23
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 coordinates 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 yaxis 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 coordinates lie within the circl
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05 Jun 2017, 09:14
Yes, it should be 45 and not 37 11,12,13,21,22,23,31,32similar points in other coordinates 8*4 01,02,03similar points in other coordinaters 3*4 00 sum 45



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Re: How many points with Integer x and Y coordinates lie within the circl
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09 Apr 2018, 06:46
When you plot the graph, the result will look as in the attachment. The radius of the circle being 4. The no of points inside the circle =3*3*4=36+(0,0)=37. Let me know if anything more needs to be explained.
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IMG_8393.JPG [ 1.38 MiB  Viewed 916 times ]



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Re: How many points with Integer x and Y coordinates lie within the circl
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12 Apr 2018, 23:20
Hello VeritasPrepKarishma / mikemcgarry, can you please give us an insight on this?



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Re: How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 00:23



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Re: How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 00:25



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Re: How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 03:09
gmatbusters, exactly! the oa is wrong. thanks for your post.



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How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 07:03
GMATinsight wrote: How many points with Integer x and Y coordinates 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) 42 D) 45 E) 49 Source: http://www.GMATinsight.comI think I should post this Solution with an apology for mentioning a wrong answer before. I didn't realize it until today so thank you all forum users for reminding me to update the OA and post solutions. I think the equation of Circle should be x^2 + y^2 = 16and the points must satisfy the expression x^2 + y^2 ≤ 16 with an equal to sign along with less than sign. This gives us 49 points in total as presented in the figure attached. Please let me know in case of any discrepancies.
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GMATINSIGHT.png [ 371.73 KiB  Viewed 747 times ]
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Re: How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 08:46
hii Thanks for the Explanation, but the points on the circle itself can't be considered as points within the circle. Or, does points within the circle include the points on the circumference ?Bunuel, chetan2u GMATinsight wrote: GMATinsight wrote: How many points with Integer x and Y coordinates 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) 42 D) 45 E) 49 Source: http://www.GMATinsight.comI think I should post this Solution with an apology for mentioning a wrong answer before. I didn't realize it until today so thank you all forum users for reminding me to update the OA and post solutions. I think the equation of Circle should be x^2 + y^2 = 16and the points must satisfy the expression x^2 + y^2 ≤ 16 with an equal to sign along with less than sign. This gives us 49 points in total as presented in the figure attached. Please let me know in case of any discrepancies.
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How many points with Integer x and Y coordinates lie within the circl
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13 Apr 2018, 10:33
gmatbusters wrote: hii Thanks for the Explanation, but the points on the circle itself can't be considered as points within the circle. Or, does points within the circle include the points on the circumference ?Bunuel, chetan2u GMATinsight wrote: GMATinsight wrote: How many points with Integer x and Y coordinates 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) 42 D) 45 E) 49 Source: http://www.GMATinsight.comI think I should post this Solution with an apology for mentioning a wrong answer before. I didn't realize it until today so thank you all forum users for reminding me to update the OA and post solutions. I think the equation of Circle should be x^2 + y^2 = 16and the points must satisfy the expression x^2 + y^2 ≤ 16 with an equal to sign along with less than sign. This gives us 49 points in total as presented in the figure attached. Please let me know in case of any discrepancies. "Point within circle" includes  Points on the circumference  Points inside the circle Both Posted from my mobile device
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Re: How many points with Integer x and Y coordinates lie within the circl
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15 Apr 2018, 05:58
afa13 wrote: Hello VeritasPrepKarishma / mikemcgarry, can you please give us an insight on this? This is my take on the question: "within the circle"  implies not beyond so I would assume we are looking at points on or inside the circle Parabola y = ax^2 + 4 ax^2 + 4 is nothing but a quadratic and since a is positive, I know it is an upward open parabola. y = ax^2 would be drawn at the origin but since we have +4 too, I will move it 4 units up. The circle has origin as the centre and a single point intersection with the parabola so it must intersect at the point (0, 4) only, the lowest point of the parabola. If it intersects at any point higher than (0, 4), it will intersect at two points. So my unique circle is ready. I will focus on one quadrant and multiply that by 4. The line y = x will divide the 1st quadrant into two equal halves. It will cut the circle at \((2\sqrt{2}, 2\sqrt{2})\) which is about (2.8, 2.8), So we know that the circle will be just shy of the (3, 3) point. So we can just draw it out and find the number of points which are on/inside it. Just draw lines across x and y coordinates which are integers. The intersection of these lines will be those points which will have both coordinates as integers. Attachment:
Diagrams.jpeg [ 21.11 KiB  Viewed 622 times ]
We see that there will be 12 such points. So we will have a total of 12*4 + 1 (the origin) = 49 points
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Re: How many points with Integer x and Y coordinates lie within the circl
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