Let a + d = c + b and a d = b c , where a , b , c ∈ R + . If the family of lines a 2 x + b 2 y + c 2 + d 2 x = 0 passes through a fixed point x 0 , y 0 , then the value of x 0 + y 0 is

Step-by-step solution

We have, a + d + 2 a d = c + b + 2 b c ⇒ a + d = c + b (As, a d = b c ) ∴ a 2 + d 2 + 2 a d = c 2 + b 2 + 2 b c ⇒ a 2 + d 2 = c 2 + b 2 Now, a 2 + d 2 x + b 2 y + c 2 = 0 ⇒ b 2 + c 2 x + b 2 y + c 2 = 0 ⇒ b 2 y + x + c 2 x + 1 = 0 ⇒ y + x + c 2 b 2 x + 1 = 0 which is of the form L 1 + λ L 2 = 0 So, the fixed point is - 1 , 1 = x 0 , y 0 (Given). Hence, x 0 + y 0 = - 1 + 1 = 0 .