If $A(6,3),B(3,5),C(4,2),P(\alpha ,\beta )$ then the ratio of the area of the triangles $PBC,ABC$ is

$\left[PBC\right]=\frac{1}{2}\left|\begin{array}{ccc}\alpha & \beta & 1\\ 3& 5& 1\\ 4& 2& 1\end{array}\right|$

$=\frac{1}{2}|\alpha \left(3\right)-\beta (-1)+1(-14)|$

$=\frac{1}{2}|3\alpha +\beta -14|$

$\left[ABC\right]=\frac{1}{2}\left|\begin{array}{ccc}6& 3& 1\\ 3& 5& 1\\ 4& 2& 1\end{array}\right|$

$=\frac{1}{2}|18-3(-1)+1(-14)|$

$=\frac{7}{2}$

$\therefore \left[PBC\right]:\left[ABC\right]=|3\alpha +\beta -14|:7$