Consider the lines L_ 1 and L_ 2 defined by L_ 1 : x 2 +y-1=0 and L_ 2 : x 2 -y+1=0 For a fixed constant , let C be the locus of a point P such that the product of the distance of P from L_ 1 and the distance of P from L_ 2 is ^ 2 . The line y=2 x+1 meets C at two points R and S, where the distance between R and S is 270 . Let the perpendicular bisector of R S meet C at two distinct points R^ and S^ . Let D be the square of the distance between R^ and S^ . The value of λ 2 is
A. A
Let the point P is h , k Distance of P from L 1 = h 2 + k - 1 ( 2 ) 2 + 1 2 = h 2 + k - 1 3 Distance of P from L 2 = h 2 - k + 1 ( 2 ) 2 + 1 2 = h 2 - k + 1 3 ∴ The equation of the locus of P is h 2 + k - 1 3 × h 2 - k + 1 3 = λ 2 h 2 + k - 1 3 h 2 - k + 1 3 = λ 2 ⇒ 2 h 2 - ( k - 1 ) 2 = 3 λ 2 Hence, the equation of the locus is 2 x 2 - ( y - 1 ) 2 = 3 λ 2 The line is y = 2 x + 1 or y - 1 = 2 x By substituting the value of y in the equation of the curve C , we get 2 x 2 - ( y - 1 ) 2 = 3 λ 2 ⇒ 2 x 2 - ( 2 x ) 2 = 3 λ 2 ⇒    2 x 2 = 3 λ 2 ⇒    x = ± 3 2 λ ⇒ x 2 - x 1 = | 6 λ | Also, y - 1 = 2 x ⇒ y 2 - 1 = 2 x 2 and y 1 - 1 = 2 x 1 ⇒ y 2 - y 1 = 2 x 2 - x 1 ⇒ y 2 - y 1 = | 2 6 λ | Given R S = 270 ⇒ x 2 - x 1 2 + y 2 - y 1 2 = 270 ⇒ ( 6 λ ) 2 + ( 2 6 λ ) 2 = 270 30 λ 2 = 270 λ 2 = 9
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