Calculus question

Question

I've tried searching the method but i cant find a clear explanation

The vertices of a triangle ΔABC are A( p,q) , B(r, s) ,C(u, v) . Assume that these

coordinates satisfy

2p + 3r + 4u = 0

2q + 3s +4v = 0

and the origin O(0,0) lies in the interior

of ΔABC . If (area of ΔOBC )=r(area of ΔABC ), then r =

(a) 2/9

(b) 4/9

(c) 2/7

(d) 3/7

(e) 4/7

Answer 1

You could do this numerically. Fix two points B and C, then find point A using the two equations given.

Then use Heron's formula on the two triangle areas : if a,b,c are three sides of a triangle

then area = sqrt (s (s-a) (s-b) (s-c)) where s = (a+b+c)/2

You can find the values of a,b,c for both triangles using the distance formula.

Then find the ratio of the areas. There's probably an algebraic solution somewhere.