Let C be the circle x ^2+( y -1)^2=2, E ₁ and E ₂ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line x+y=3 touch the curves C , E₁ and E₂ at P (x₁, y₁ ), Q (x₂, y₂ ) and R (x₃, y₃ ) respectively. Given that P is the mid-point of the line segment Q R and P Q= 2 2 3 , the value of 9 (x₁ y₁+x₂ y₂+x₃ y₃ ) is equal to ______ .

Step-by-step solution

Let E ₁: x ^2 a ^2 + y ^2 ~b ^2 =1,( a b ) E ₂: x ^2 c ^2 + y ^2 ~d ^2 =1,( c d ) C: x^2+(y-1)^2=2 Equation of tangent at P ( x ₁, y ₁ ) xx ₁+ y ( y ₁-1 )= ( y ₁+1 ) comparing with x + y =3 we get P (1,2) Now parametric equation of x+y=3 ( x -1) ( -1 2 ) = ( y -2) ( 1 2 ) = 2 2 3 ( PQ = 2 2 3 ) On solving we get Q ( 5 3 , 4 3 ), R ( 1 3 , 8 3 ) So, 9 ( x ₁ y ₁+ x ₂ y ₂+ x ₃ y ₃ ) aligned & 9 (2+ 5 3 4 3 + 1 3 8 3 ) & 46 aligned