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!
Sinchan Dutta Chaudhuri , 6 Years ago
Grade 11
Analytical Geometry> Please Find the number of integral points...
3 AnswersAditya Gupta
Last Activity: 6 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
Last Activity: 6 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
Last Activity: 6 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
Provide a better Answer & Earn Cool Goodies
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Ask a Doubt
Get your questions answered by the expert for free
Other Related Questions on analytical geometry
- The locus of the point represented by x= t^2 +t+1 and y=t^-t+1 is
- The locus of the point represented by x= t^2 +t+1 and y=t^-t+1 is
analytical geometry
0 Answer Available
Last Activity: 2 Years ago
show that the centroids of the triangles of which three perpendiculars lie along the lines y=m1x,y=m2x,y=m3x lie on y(3+m2m3+m3m2+m1m2)=x(m1+m2+m3+3m1m2m3)
show that the centroids of the triangles of which three perpendiculars lie along the lines y=m1x,y=m2x,y=m3x lie on y(3+m2m3+m3m2+m1m2)=x(m1+m2+m3+3m1m2m3)
analytical geometry
0 Answer Available
Last Activity: 2 Years ago
show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0
show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0
analytical geometry
0 Answer Available
Last Activity: 2 Years ago
A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius.
A circle has its center on straight line 5x-2y+1=0 and cuts the x axis at two points whose absciasse are -5,3 find equation of the circle and it's radius.
analytical geometry
0 Answer Available
Last Activity: 3 Years ago
Prove that ax^2+by^2+cz^2 +2ux+2vy+2wz+d=0 represent a cone if u^2/a+v^/b+w^2/c=d
Prove that ax^2+by^2+cz^2 +2ux+2vy+2wz+d=0 represent a cone if u^2/a+v^/b+w^2/c=d
analytical geometry
0 Answer Available
Last Activity: 3 Years ago