0Question:
How many pairs of integers (x, y) exist such that x^2+ 4*y^2<= 100?
Solution:
Method 1:
Case and Fundamental counting:
y = 0, 1, 2 =>19 ways
y = 3 =>17ways
y = 4 =>13 ways
y = 5 => 1 way
Total = 19*3+17+13+1=88
Similarly for y negtve total ways =19*2+17+13+1=69
Again for y=10 x=0 and y=-10 x=0
So total ways = 88+69+2=159
Method 2:
Ellipse: Geometrical approach:
x^2/10^2 + y^2/5^2 = 1 => Ellipse
Case 1: 4 Boundary points cutting x and y axis.
Case 2: x =+/- 9 => y = +/-2 and 0 => 19*5 = 95 points
Case 3: y = +/- 3 => x^2 <= 64 => 2 * 17 = 34 points
Case 4: y = +/- 4 => x^2 <= 36 => 2 * 13 = 26 points
Total: 4 + 95 + 60 = 159 points
