If the angle between the circles x 2 + y 2 - 12 x - 6 y + 41 = 0 and x 2 + y 2 + k x + 6 y - 59 = 0 is 45 ° , then a value of k is

Step-by-step solution

We know that angle between two circles x 2 + y 2 + 2 g 1 x + 2 f 1 y + c ' = 0 and x 2 + y 2 + 2 g 2 x + 2 f 2 y + c ' ' = 0 is given by cos θ = c 1 c 2 2 - r 1 2 - r 2 2 2 r 1 r 2 where, c 1 c 2 is the distance between the centres of the two circles and r 1   &   r 2 are the radius of the circles. Now, C 1 ≡ x 2 + y 2 - 12 x - 6 y + 41 = 0 Here, g 1 = - 6 ,   f 1 = - 3 ,   c ' = 41 Therefore, c 1 ≡ 6 , 3 r 1 = g 1 2 + f 1 2 - c ' = 36 + 9 - 41 = 2 And, C