Geo3.6

Find the centroid of the triangle formed by the lines $x^2-5xy+4y^2=0\,$ and $x+2y-6=0\,$

Given pair of lines $x^2-5xy+4y^2=0\,$

$(x-y)(x-4y)=0\,$

Therefore the three sides of the triangle are $x=y,x-4y=0,x+2y-6=0\,$

Equations 1 and 2 meet at the origin.

Solving 1 and 3, we get $A(2,2)\,$

Solving 2 and 3,we get $B(4,1)\,$

Therefore,the centroid of the triangle is $\left(\frac{2+4}{3},\frac{2+1}{3}\right)\,$

The centroid is $(2,1)\,$