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How many points with integer coordinates lie on the circumference of c
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06 Sep 2023, 09:19
06 Sep 2023, 09:19
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How many points with integer coordinates lie on the circumference of circle \(x^2 + y^2 = 5^2\)?
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Re: How many points with integer coordinates lie on the circumference of c
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18 Sep 2023, 08:12
18 Sep 2023, 08:12
Expert Reply
OE
Equation of a circle, having origin as its centre, is in the form
๐ฅ^2 + ๐ฆ^2 = ๐ ^2
Where ๐ is the radius of the circle.
Given equation is ๐ฅ^2 + ๐ฆ^2 = 5^2
That is ๐ฅ^2 + ๐ฆ^2 = 25,
Itโs a circle with a centre at origin and radius is equal to 5.
Let us check, for which all values of ๐ฅ and ๐ฆ in the first quadrant, the equation gets
satisfied.
We can observe, ๐ฅ ranges from 0 to 5.
If ๐ฅ = 0, then ๐ฆ = 5.
If ๐ฅ = 1, then ๐ฆ = โ24 , as โ24 is not an integer so we will not consider this case
.so we cannot put ๐ฅ as 1.
If we put ๐ฅ = 2, then ๐ฆ = โ21 , as โ21 is not an integer so we will not consider
this case also.
If we put ๐ฅ = 3 then ๐ฆ = 4,
As ๐ฅ and ๐ฆ both are integers so will consider this case.
So, we have (3,4) a point, which satisfy equation of circle as 3^2 + 4^2 = 25
If we put ๐ฅ = 4 then ๐ฆ = 3,
As ๐ฅ and ๐ฆ both are integers so will consider this case, also.
So, we have (4,3) a point, which satisfy equation of the circle as, 4^2 + 3^2 = 25
So, in the first quadrant we have two points which satisfy the equation of the circle.
Similarly, weโll have two points in the second quadrant. As (- 3, 4) and (-4, 3).
Weโll have two points in the third quadrant. As (- 3, - 4) and (-4, -3)
And weโll have two points in the fourth quadrant. As (3, -4) and (4, -3).
So, we have 8 points which satisfy the equation of line.
Now let us consider the points where the circle crosses ๐ฅ and ๐ฆ โaxis.
Points (5,0), (0,5), (-5,0) and (0, -5) will satisfy the equation of line.
So, in total we have 12 points satisfying the equation of line.
Ans. (12)
Equation of a circle, having origin as its centre, is in the form
๐ฅ^2 + ๐ฆ^2 = ๐ ^2
Where ๐ is the radius of the circle.
Given equation is ๐ฅ^2 + ๐ฆ^2 = 5^2
That is ๐ฅ^2 + ๐ฆ^2 = 25,
Itโs a circle with a centre at origin and radius is equal to 5.
Let us check, for which all values of ๐ฅ and ๐ฆ in the first quadrant, the equation gets
satisfied.
We can observe, ๐ฅ ranges from 0 to 5.
If ๐ฅ = 0, then ๐ฆ = 5.
If ๐ฅ = 1, then ๐ฆ = โ24 , as โ24 is not an integer so we will not consider this case
.so we cannot put ๐ฅ as 1.
If we put ๐ฅ = 2, then ๐ฆ = โ21 , as โ21 is not an integer so we will not consider
this case also.
If we put ๐ฅ = 3 then ๐ฆ = 4,
As ๐ฅ and ๐ฆ both are integers so will consider this case.
So, we have (3,4) a point, which satisfy equation of circle as 3^2 + 4^2 = 25
If we put ๐ฅ = 4 then ๐ฆ = 3,
As ๐ฅ and ๐ฆ both are integers so will consider this case, also.
So, we have (4,3) a point, which satisfy equation of the circle as, 4^2 + 3^2 = 25
So, in the first quadrant we have two points which satisfy the equation of the circle.
Similarly, weโll have two points in the second quadrant. As (- 3, 4) and (-4, 3).
Weโll have two points in the third quadrant. As (- 3, - 4) and (-4, -3)
And weโll have two points in the fourth quadrant. As (3, -4) and (4, -3).
So, we have 8 points which satisfy the equation of line.
Now let us consider the points where the circle crosses ๐ฅ and ๐ฆ โaxis.
Points (5,0), (0,5), (-5,0) and (0, -5) will satisfy the equation of line.
So, in total we have 12 points satisfying the equation of line.
Ans. (12)
Re: How many points with integer coordinates lie on the circumference of c
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29 Oct 2024, 19:02
29 Oct 2024, 19:02
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Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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