Let the equation x 2 + y 2 + p x + ( 1 - p ) y + 5 = 0 represent circles of varying radius r ∈ ( 0 , 5 ] . Then the number of elements in the set S = q : q = p 2 and q is an integer is ___________

Step-by-step solution

r = p 2 4 + 1 - p 2 4 - 5 = 2 p 2 - 2 p - 19 2 Since r ∈ ( 0 , 5 ] So, 0 < 2 p 2 - 2 p - 19 2 ≤ 5 0 < 2 p 2 - 2 p - 19 ≤ 100 ⇒ 2 p 2 - 2 p - 19 > 0       . . . i For, 2 p 2 - 2 p - 19 > 0 p = 2 ± 4 - 4 × 2 × - 19 4 = 2 ± 156 4 ⇒ p - 1 + 39 2 p - 1 - 39 2 > 0 . . . . i i p 2 = 10 ± 39 2 Also, 2 p 2 - 2 p - 119 ≤ 0       . . . i i i For, 2 p 2 - 2 p - 119 = 0 p = 2 ± 4 - 4 × 2 × - 119 4 = 1 &#