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Q.
Three points are $A(6,3), B(-3,5), C(4,-2)$ and $P(x, y)$ is any point, then the ratio of area of $\triangle P B C$ and $\Delta A B C$ is :
Bihar CECEBihar CECE 2003
Solution:
If $A\left(x_{1},\, y_{1}\right),\, B\left(x_{2},\, y_{2}\right)$
and $C\left(x_{3},\, y_{3}\right)$ are the vertices of a triangle, then Area of
$\Delta=\frac{1}{2}\left\{x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)\right.$
$\left. +x_{3}\left(y_{1}-y_{2}\right)\right\}$
Given points are $A(6,3), B(-3,5), C(4,-2)$ and $P(x, y)$
$\therefore \frac{\Delta P B C}{\Delta A B C}$
$=\frac{\frac{1}{2}}{\frac{1}{2}}\left[\frac{x(5+2)-3(-2-y)+4(y-5)}{6(5+2)-3(-2-3)+4(3-5)}\right]$
$=\left[\frac{7 x+7 y-14}{42+15-8}\right]$
$=\frac{7 x+7 y-14}{49}=\frac{x+y-2}{7}$