If ( m _ i , 1 ~m _ i ), m _ i 0, i =1,2,3,4 are four distinct points on a circle, then the product m ₁ ~m ₂ ~m ₃ ~m ₄ is equal to

Step-by-step solution

Let (m_i, 1 m_i ) lie on the circle x^2+y^2+2 ~g x+2 f y+ c =0 where i =1,2,3,4 aligned & (m_i )^2+ ( 1 m_i )^2+2 gm _ i + 2 f m_i + c =0 & ~m _ i ^2+ 1 ~m _ i ^2 +2 gm _ i + 2 f ~m _ i + c =0 & ~m _ i ^4+1+2 gm _ i ^3+2 fm _ i + cm _ i ^2+2=0 & ~m _ i ^4+2 gm _ i ^3+ cm _ i ^2+2 fm _ i +1=0 aligned m ₁, ~m ₂, ~m ₃, ~m ₄ are roots of the above equation product of roots = e a = 1 1 =1 m ₁ ~m ₂ ~m ₃ ~m ₄=1