If a variable line drawn through the intersection of the lines x 3 + y 4 = 1 and x 4 + y 3 = 1 , meets the coordinate axes at A and B , A ≠ B , then the locus of the midpoint of AB is:

Question

If a variable line drawn through the intersection of the lines x 3 + y 4 = 1 and x 4 + y 3 = 1 , meets the coordinate axes at A and B , A ≠ B , then the locus of the midpoint of AB is:, 7 x y = 6 x + y, 4 x + y 2 - 28 x + y + 49 = 0, 6 x y = 7 x + y, 14 x + y 2 - 97 x + y + 168 = 0

MARKS App · Accepted Answer

L 1 : 4 x + 3 y - 12 = 0 L 2 : 3 x + 4 y - 12 = 0 L 1 + λ L 2 = 0 ⇒ 4 x + 3 y - 12 + λ 3 x + 4 y - 12 = 0 ⇒ x 4 + 3 λ + y 3 + 4 λ - 12 1 + λ = 0 Point A 12 1 + λ 4 + 3 λ , 0 Point B 0 , 12 1 + λ 3 + 4 λ Mid point ⇒ h = 6 1 + λ 4 + 3 λ ....... i k = 6 1 + λ 3 + 4 λ ....... ii Eliminate λ from equations i and ii then 6 h + k = 7 h k Hence, locus is 6 x + y = 7 x y

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