Question

Find the co - ordinates of the centroid and circumcentre and orhocentre of the triangle formed by the lines 3x - 2y - 6 = 0, 3x + 4y + 12 = 0 and 3x - 8y + 12 = 0

Answer 1

Solving in pairs the co - ordinate of vertices of the triangle are $A(0,3),B(4,3),C(-4,0)$ (A,C,B order according to equation ordering)

If G be the centroid, then

$G=(\frac{\sum x}{-3},\frac{\sum y}{3})=(0,0)$ ....(1)
If O be the circumcentre , then OA = OB = OC gives

$2x+3y-4=0and8x-6y+7=0$
Solving the above, point O is $(\frac{1}{2},\frac{23}{18})$

If $H(\alpha ,\beta )$ be the orthocentre , then G divides OH in the ratio 1:2.

$0=\frac{\alpha +2.\left(1/12\right)}{1+2}and\frac{\beta +2.\left(23/18\right)}{1+2}$

$\therefore (\alpha ,\beta )is(-\frac{1}{6},-\frac{23}{9})$