Let the point p , p + 1 lie inside the region E = x , y : 3 - x ≤ y ≤ 9 - x 2 , 0 ≤ x ≤ 3 . If the set of all values of p is the interval a , b , then b 2 + b - a 2 is equal to _ _ _ _ _ _ _ _ .

Step-by-step solution

Given, The point p , p + 1 lie inside the region E = x , y : 3 - x ≤ y ≤ 9 - x 2 , 0 ≤ x ≤ 3 , And the set of all values of p is the interval a , b , Now by observation p , p + 1 lies on y - x = 1 , Also point p , p + 1 lies on x + y = 3 So, solving y - x = 1 and x + y = 3 we get, P 1 , 2 Again given point p , p + 1 lies on x 2 + y 2 = 9 So, solving y - x = 1 and x 2 + y 2 = 9 , we get x 2 + ( 1 + x ) 2 = 9 ⇒ 2 x 2 + 2 x - 8 = 0 ⇒ x 2 + x - 4 = 0 . . . . . . . . . i ⇒ x = -