Question

∆OAB is formed by lines x² – 4xy + y² = 0 and the line 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 \(\left(\cfrac{x_1+x_2}{2},\cfrac{y_1+y_2}{2}\right)\)

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 = \(\cfrac{1-3y}{2}\) in the equation x² – 4xy + y² = 0,

we get,

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

Since D lies on the line AB, we can find the xcoordinate of D as