Straight Line and Pair of Straight Lines 3 Question 12
13. The coordinates of $A, B, C$ are $(6,3),(-3,5),(4,-2)$ respectively and $P$ is any point $(x, y)$. Show that the ratio of the areas of the triangles $\triangle P B C$ and $\triangle A B C$ is $\left|\frac{x+y-2}{7}\right|$.
$(1983,2 M)$
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Solution:
- Area of $\triangle P B C=\frac{\frac{1}{2}|x(5+2)+(-3)(-2-y)+4(y-5)|}{\frac{1}{2}|6(5+2)+(-3)(-2-3)+4(3-5)|}$
$$ =\frac{|7 x+7 y-14|}{|42+15-8|}=\frac{7|x+y-2|}{49}=\frac{x+y-2}{7} $$
