Find the number of integral points on the co-ordinate axes which are at a distance
k units (0≤k≤3,) from the point (2,3)
Answers
Given : points on the co-ordinate axes which are at a distance
k units (0≤k≤3,) from the point (2,3)
To find : number of integral points
Solution:
We can draw a circle with center ( 2, 3) & Radius = 3
All points lying inside will meet the required Criteria
4 points ( 2 , 6 ) ( 2 , 0) & ( 5 , 3) & ( -1 , 3)
Boundaries of x are -1 to 5
boundaries of y are 0 to 6
(x - 2)² + ( y - 3)² ≤ 3²
x = -1
( y - 3)² ≤ 0
=> y = 3
x = 0
( y - 3)² ≤ 5
=> y = 5 , 4 , 3 , 2 , 1
x = 1
( y - 3)² ≤ 8
=> y = 5 , 4 , 3 , 2 , 1
x = 2
( y - 3)² ≤ 9
=> y = 6, 5 , 4 , 3 , 2 , 1 , 0
x = 3
( y - 3)² ≤ 8
=> y = 5 , 4 , 3 , 2 , 1
x = 4
( y - 3)² ≤ 5
=> y = 5 , 4 , 3 , 2 , 1
x = 5
( y - 3)² ≤ 0
=> y = 3
1 + 5 + 5 + 7 + 5 + 5 + 1 = 29 points
number of integral points on the co-ordinate axes which are at a distance
k units (0≤k≤3,) from the point (2,3) are 29
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