Area of the region x , y ∈ R 2 : y ≥ x + 3 , 5 y ≤ x + 9 ≤ 15 is equal to

Step-by-step solution

If x + 3 0 ⇒ y ≥ − ( x + 3 ) & y ≥ 0 If x + 3 ≥ 0 ⇒ y ≥ ( x + 3 ) & y ≥ 0 Also 5 y ≤ x + 9 ≤ 15 ⇒ x − 5 y + 9 ≥ 0 & x ≤ 6 Intersection point of line & parabolas 5 y = x + 9 & y 2 = | x + 3 | ( y ≥ 0 ) ( x + 9 5 ) 2 = | x + 3 | x 2 + 18 x + 81 = 25 x + 75 o r − 25 x − 75 x 2 − 7 x + 6 = 0 o r x 2 + 43 x + 156 = 0 x = 1 , 6 o r x = − 39 , − 4 So Clearly, x = 1 , y = 2 and when x = − 4 , y = 1 are required points Drawing all these, we get this graph. Shaded portion is required area. Clearly required area = area (trape