If the distance between the lines represented by (x - 2y)^2 + k(x - 2y) = 0 is 3 units and k > 0 then the equations of the lines are

Step-by-step solution

The given equation is (x - 2y)^2 + k(x - 2y) = 0 . Factoring the equation, we get: (x - 2y)(x - 2y + k) = 0 This represents two parallel lines: x - 2y = 0 x - 2y + k = 0 The distance between two parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0 is given by |c₁ - c₂| a^2 + b^2 . Applying this formula, the distance between the lines is: d = |k - 0| 1^2 + (-2)^2 = |k| 5 Given that the distance is 3 units: |k| 5 = 3 |k| = 3 5 Since k > 0 , we have k = 3 5 . Substituting the value of k , the equations of the lines ar