If the normals drawn to the hyperbola (x y=4 ) at ( ( _i, _i )(i=1,2,3,4) ) are concurrent at the point ((a, b) ), then ( ( ₁+ ₂+ ₃+ ₄ ) ( ₁+ ₂+ ₃+ ₄ ) ( ₁ ₂ ₃ ₄ )= )

Step-by-step solution

The equation of normal to the given hyperbola (x y=4 ) at point ( (2 t, 2 t ) ) is (2 t^4-x t^3+y t-2=0 ) ...(i) ( ) Normal (i) passes through point ((a, b) ), so (2 t^4-a t^3+b t-2=0 ), the equation having roots are ( ₁ 2 , ₂ 2 , ₃ 2 ) and ( ₄ 2 ), so ( gathered 1 2 ( ₁+ ₂+ ₃+ ₄ )= a 2 ₁+ ₂+ ₃+ ₄=a ( ₁ ₂ 4 )=0 ₁ ₂ ₃ 8 = -b 2 and ₁ ₂ ₃ ₄ 16 =-1 ₁+ ₂+ ₃+ ₄= 4 ₁ + 4 ₂ + 4 ₃ + 4 ₄ gathered ) (=4 ₁ ₂ ₃ ₁ ₂ ₃ ₄ ) (=4 -b / 2 8 -16 =b ) ( aligned . ₁+ ₂+ ₃+ ₄ ) ₁+ ₂+ ₃+ ₄ & ( ₁ ₂ ₃ ₄ ) = & a(-16) b =-16 a b aligned ) Henc