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 ______.

Options

• 7xy = 6(x + y)

• 4(x + y)² – 28(x + y) + 49 = 0

• 6xy = 7(x + y)

• 14(x + y)² – 97(x + y) + 168 = 0

Answer 1

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 7xy = 6(x + y).

Explanation:

L₁ : 4x + 3y – 12 = 0

L₂ : 3x + 4y – 12 = 0

L₁ + λL₂ = 0

(4x + 3y – 12) + λ(3x + 4y – 12) = 0

x(4 + 3λ) + y(3 + 4λ) – 12(1 + λ) = 0

Point A `((12(1 + λ))/(4 + 3λ), 0)`

Point B `(0, (12(1 + λ))/(3 + 4λ))`

Mid point ⇒ h = `(6(1 + λ))/(4 + 3λ)`  ....(i)

k = `(6(1 + λ))/(3 + 4λ)`  ....(ii)

Eliminate λ from (i) and (ii), then

6(h + k) = 7hk

6(x + y) = 7xy