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# How many points on the circumference of a semi-circle

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How many points on the circumference of a semi-circle [#permalink]  27 Jun 2008, 06:46
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How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 08:14
x^2+y^2=5 , so it is a circle with (0,0) as its orgin, with radius equal to 5^0.5 = 2.2 approx.

The 4 vertix (typo?) of the circles are exactly the 4 intercepts with the x-, y- axis.

So the 4 vertix are (2.2,0), (0,-2.2), (-2.2,0), and (0,2.2)

Let's go through the circumference, starting from (2.2,0), in clockwise direction:
the first point on the circumference with integer coordinates is:
(2,-1)
(1,-2)
(-1,-2)
(-2,-1)
(-2,1)
(-1,2)
(1,2)
(2,1)

Current Student
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 08:20
What do you call a "semi circle"?
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 08:27
Oski wrote:
What do you call a "semi circle"?

a half-circle.. could be in the qudarant 1 & 2.. or 3&4..or 2&3 or 1&4..

however key is to look at on 2 quadarants..doesnt matter the orientation of the semi-circle
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 09:00
Its A! for any semi circular region and 8 for whole circle!!
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 10:16
fresinha12 wrote:
How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted

Assuming semi circle goes through quadrant I and II -
If y = 1, x = -2 or +2
y = 2, x = -1, +1

If x = -1 or +1, y = 2, since its a semi circle (positive y axis only )
If x = -2 or +2, y = 1,

thats comes to 6
Ans :B.
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Re: Coordinate Circle [#permalink]  27 Jun 2008, 19:43
judokan wrote:
x^2+y^2=5 , so it is a circle with (0,0) as its orgin, with radius equal to 5^0.5 = 2.2 approx.

The 4 vertix (typo?) of the circles are exactly the 4 intercepts with the x-, y- axis.

So the 4 vertix are (2.2,0), (0,-2.2), (-2.2,0), and (0,2.2)

Let's go through the circumference, starting from (2.2,0), in clockwise direction:
the first point on the circumference with integer coordinates is:
(2,-1)
(1,-2)
(-1,-2)
(-2,-1)
(-2,1)
(-1,2)
(1,2)
(2,1)

Sorry, I overlook the question, it is concern about semi-circle.

My answer would be 4 (A)
(2,-1)
(1,-2)
(-1,-2)
(-2,-1)
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Re: Coordinate Circle [#permalink]  02 Jul 2008, 03:08
The answer is A, 4 points- two for every quarter
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Re: Coordinate Circle [#permalink]  02 Jul 2008, 08:06
x^2+y^2=5

=> x and y can have values |2| or |1|

we can have eight such values.

2,1
-2,1
2,-1
-2,-1

and

1,2
-1,2
1,-2
-1,-2
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 02:56
fresinha12 wrote:
How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted

I have an issue with your problem. As long as the diameter of the semi-circle is parallel to an axis, the answer is 4, but it is possible to define semi-circles centered on the origin that pass through 5 integer coordinates ( e.g. m = -1/2 ).
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 03:41
iamcartic wrote:
Its A! for any semi circular region and 8 for whole circle!!

Is this a general rule somewhere?
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 04:38
Permuteer wrote:
fresinha12 wrote:
How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted

I have an issue with your problem. As long as the diameter of the semi-circle is parallel to an axis, the answer is 4, but it is possible to define semi-circles centered on the origin that pass through 5 integer coordinates ( e.g. m = -1/2 ).

But 5 is not a given option. Unless you want to pass the question, u have to take the one that makes most sense.
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 05:42
fresinha12 wrote:
How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted

Since this is a semi circle with origin at (0,0) , integer co-oridnates are (assuming semi-circle lies on x-axis)

(1,2)
(-1,2)
(2,1)
(-2,1)
(-2,0)
(-1,0)
(1,0)
(2,0)

Number of co-ordinates are 8.

Remember circumference of semi-circle = r(pi + 2)
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 06:03
nmohindru wrote:
Since this is a semi circle with origin at (0,0) , integer co-oridnates are (assuming semi-circle lies on x-axis)

(1,2)
(-1,2)
(2,1)
(-2,1)
(-2,0)
(-1,0)
(1,0)
(2,0)

Number of co-ordinates are 8.

Remember circumference of semi-circle = r(pi + 2)

An interesting approach. I interpreted the "circumference of a semi-circle" as referring only to the curved part of the figure. I wonder if there is a rule for it. Still, wouldn't your interpretation yield nine points when (0,0) is counted?
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 06:05

(-2, 0) or (-1,0) is not a solution. Just put in the equation, you will never get 5 as a result.
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 07:44
I would have never caught the semi-circle catch in the question but yeah agree with you that answer should be 4.
with 8c4 = 70 different possibilities..
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 08:02
zoltan wrote:

(-2, 0) or (-1,0) is not a solution. Just put in the equation, you will never get 5 as a result.

You might be right but Q does not seem to be proper to me
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 08:54
Permuteer wrote:
I have an issue with your problem. As long as the diameter of the semi-circle is parallel to an axis, the answer is 4, but it is
possible to define semi-circles centered on the origin that pass through 5 integer coordinates ( e.g. m = -1/2 ).

could you expand on that?
what is m?
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 17:12
sset009 wrote:
Permuteer wrote:
I have an issue with your problem. As long as the diameter of the semi-circle is parallel to an axis, the answer is 4, but it is
possible to define semi-circles centered on the origin that pass through 5 integer coordinates ( e.g. m = -1/2 ).

could you expand on that?
what is m?

"m" is the common abbreviation for slope.

1) plot all eight answers for the original equation
2) draw circle through all eight points.
3) draw a line from (-2,1) to (2,-1). This line will pass through the origin and will have a slope of -1/2.
4) consider this line the diameter of the semi-circle.
5) the semi circle now passes through 5 points: (-2,1) (-1,2) (1,2) (2,1) (2,-1)
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Re: Coordinate Circle [#permalink]  22 Jul 2008, 17:22
fresinha12 wrote:
How many points on the circumference of a semi-circle represented with x^2+y^2=5 have integer coordinates?
(A) 4
(B) 6
(C) 8
(D) 12
(E) 16

just thought i'd present a lil twist to a problem ritula posted

A: (-2,-1),(-1,-2), (1,2), (2,1)
Re: Coordinate Circle   [#permalink] 22 Jul 2008, 17:22
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