Question

ΔOAB is formed by the lines x² - 4xy + y² = 0 and the line AB. The equation of line AB is 2x + 3y - 1 = 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 2x + 3y - 1 = 0

∴ points A and B satisfy the equations 2x + 3y - 1 = 0 and x² - 4xy + y² = 0 simultaneously. 

We eliminate x from the above equations, i.e., put x = `(1 - "3y")/2` in the equation x² - 4xy + y² = 0, we get, 

∴ `((1 - 3"y")/2)^2 - 4 ((1 - "3y")/2)"y" + "y"^2 = 0`

∴ `(1 - "3y")^2 - 8(1 - "3y")"y" + "4y"^2 = 0`

∴ 1 - 6y + 9y² - 8y + 24y² + 4y² = 0 

∴ 37y² - 14y + 1 = 0

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

∴ y₁ + y₂ = `-"b"/"a" = 14/37`

∴ y-coordinate of D = `("y"_1 + "y"_2)/2 = 7/37`

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

2x + 3`(7/37) - 1 = 0`

∴ 2x = `1 - 21/37 = 16/37`

∴ x = `8/37`

∴ D is `(8/37, 7/37)`

∴ equation of the median OD is `"x"/(8//37) = "y"/(7//37)`

i.e. 7x - 8y = 0