Question

ΔOAB is formed by lines x² – 4xy + y² = 0 and the line x + y – 2 = 0. Find the equation of the median of the triangle drawn from O.

Answer 1

Let D be the midpoint of seg AB, where A is (x₁, y₁) and B is (x₂, y₂).

Then D has coordinates `((x_1 + x_2)/2, (y_1 + y_2)/2)`.

The joint (combined) equation of the lines OA and OB is x² – 4xy + y² = 0 and the equation of the line AB is x + y – 2 = 0.

∴ Points A and B satisfy the equations x + y – 2 = 0 and x² – 4xy + y² = 0 simultaneously.

We eliminate x from the above equations.

Put x = 2 – y in the equation x² – 4xy + y² = 0, we get

(2 – y)² – 4(2 – y)y + y² = 0

∴ 4 – 4y + y² – 8y + 4y² + y² = 0

∴ 6y² – 12y + 4 = 0

∴ 3y² – 6y + 2 = 0

The roots y₁ and y₂ of this quadratic equation are the y-coordinates of the points A and B.

∴ y₁ + y₂ = `-b/a = 6/3` = 2

∴ y-coordinate of D = `(y_1 + y_2)/2 = 2/2` = 1

Since D lies on the line AB, we can find the x-coordinate of D as

x + 1 – 2 = 0

∴ x = 1

∴ D is (1, 1)

∴ Equation of the median OD is `(y - 0)/(x - 0) = (1 - 0)/(1 - 0)` = 1

∴ y = x,

i.e. x – y = 0