Total number of lines of the form p x + q y + r = 0 (where p q ≠ 0 ) that intersect the ellipse x 2 2 + y 2 = 50 at two integral points (points whose coordinates are integers) is equal to

Step-by-step solution

If y ∈ I then y ∈ - 7 , - 6 , . . . . , 0,1 , . . . , 6,7 Total integral points on x 2 2 + y 2 = 50 are 12 which are 2,7 , 14,1 , 2 , - 7 - 14,1 , - 2,7 , 14 , - 1 - 2 , - 7 , - 14 , - 1 , 10,5 10 , - 5 , - 10,5 - 10 , - 5 Total number of lines passing through at least two points among these are 12 C 2 = 66 in which x = 2 , x = - 2 ,   x = 14 ,   x = - 14 ,   x = 10 ,   x = - 10 , y = 1 ,   y = 7 ,   y = - 1 ,   y = - 7 ,   y = 5 ,   y = - 5 are rejected. &#8