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
Enter text here...
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
Enter text here...
Other Related Questions on analytical geometry
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)
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
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.
analytical geometry
0 Answer Available
Last Activity: 2 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
analytical geometry
0 Answer Available
Last Activity: 2 Years ago