If the angle between the circles x 2 + y 2 + 4 x - 5 = 0 and x 2 + y 2 + 2 λ y - 4 = 0 is π 3 then λ =

Step-by-step solution

The equation of the circles is given as,  x 2 + y 2 + 4 x - 5 = 0 x 2 + y 2 + 2 λ y - 4 = 0 On comparing in equations with  x 2 + y 2 + 2 g x + 2 f y + c = 0 g 1 = 2 , f 1 = 0 , c 1 = - 5 g 2 = 0 , f 2 = λ , c 2 = - 4 The angle between the circles is,  cos θ = 2 g 1 g 2 + 2 f 1 f 2 - c 1 - c 2 2 g 1 2 + f 1 2 - c 1 g 2 2 + f 2 2 - c 2 cos π 3 = 2 × 2 × 0 + 2 × 0 × λ + 5 + 4 2 4 + 0 + 5 0 + λ 2 + 4 1 2 = 3 2 λ 2 + 4 After simplification,  &#9