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Please Find the number of integral points lying within the circle: X^2+Y^2=100 Thanks!

Please Find the number of integral points lying within the circle: X^2+Y^2=100 Thanks!
 

Grade:11

3 Answers

Aditya Gupta
2075 Points
3 years ago
actually, the questions you have asked above is a legendary question in discrete mathematics (combinatorics in particular).
we write x^2=100-y^2=(10-y)(10+y).
we now check each value of y from 0 to 10
y=0 and 10 work (easy)
similarly, y=6 and 8 work, while others dont.
so, in the first quadrant, we got (8,6), (6,8) satisfying the given criteria.
in second quad, we have (-8,6), (-6,8)
third, we got (-8,-6), (-6,-8)
and in fourth, (8,-6), (6,-8)
also, on the axes, we have (0,10), (0,-10), (10,0), (-10,0)
so in total, we have 12 points, but the points on the axes are not within the circle, they lie on the circle.
So, we have 8 integral points within the circle.
Aditya Gupta
2075 Points
3 years ago
sorry for a wrong answer above :(.
we see that in first quadrant, when x=1,2,3,4, y will attain all values from 1 to 9 because 9^2+4^2
when x=5, y can attain all values from 1 to 8
when x=6,7 y can attain all values from 1 to 7
when x=8, y can attain all values from 1 to 5
when x=9, y can attain all values from 1 to 4. so total number of points in first quadrant are 9*4+8+2*7+5+4=67
so points in all quadrants=67*4=268
now, number of points on x axis are 9.
so points on all axes are 9*4=36 
including the origin, we get total points as 268+9+1=278
 
 
 
Aditya Gupta
2075 Points
3 years ago
 
sorry for a wrong answer above :(.
we see that in first quadrant, when x=1,2,3,4, y will attain all values from 1 to 9 because 9^2+4^2
when x=5, y can attain all values from 1 to 8
when x=6,7 y can attain all values from 1 to 7
when x=8, y can attain all values from 1 to 5
when x=9, y can attain all values from 1 to 4. so total number of points in first quadrant are 9*4+8+2*7+5+4=67
so points in all quadrants=67*4=268
now, number of points on x axis are 9.
so points on all axes are 9*4=36 
including the origin, we get total points as 268+36+1=305
 

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