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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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New post Updated on: 13 Apr 2018, 06:53
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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) 42
D) 45
E) 49

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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 co-ordinates lie within the circl  [#permalink]

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New post 13 May 2017, 06:33
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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, 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 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 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 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, 09:14
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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Re: How many points with Integer x and Y co-ordinates lie within the circl  [#permalink]

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New post 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
IMG_8393.JPG [ 1.38 MiB | Viewed 723 times ]

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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 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 co-ordinates lie within the circl  [#permalink]

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New post 13 Apr 2018, 00:23
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Hii

Please see my attached sketch.
Feel free to ask again if you have any more doubts.

afa13 wrote:
Hello VeritasPrepKarishma / mikemcgarry,
can you please give us an insight on this?

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IMG-20180413-WA0029.jpg
IMG-20180413-WA0029.jpg [ 97.75 KiB | Viewed 613 times ]


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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 Apr 2018, 00:25
Please recheck the OA.

GMATinsight 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

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

Source: http://www.GMATinsight.com

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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 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 co-ordinates lie within the circl  [#permalink]

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New post 13 Apr 2018, 07:03
GMATinsight 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

A) 16
B) 17
C) 42
D) 45
E) 49

Source: http://www.GMATinsight.com


I 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 = 16

and 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.
Attachments

File comment: www.GMATinsight.com
GMATINSIGHT.png
GMATINSIGHT.png [ 371.73 KiB | Viewed 555 times ]


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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 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 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) 42
D) 45
E) 49

Source: http://www.GMATinsight.com


I 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 = 16

and 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 co-ordinates lie within the circl  [#permalink]

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New post 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 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) 42
D) 45
E) 49

Source: http://www.GMATinsight.com


I 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 = 16

and 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 co-ordinates lie within the circl  [#permalink]

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New post 15 Apr 2018, 05:58
1
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 co-ordinates which are integers. The intersection of these lines will be those points which will have both co-ordinates as integers.
Attachment:
Diagrams.jpeg
Diagrams.jpeg [ 21.11 KiB | Viewed 430 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 co-ordinates lie within the circl  [#permalink]

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