Question

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 $△PBC$ and $\Delta ABC$ is :

• A. $\frac{x+y-2}{7}$
• B. $\frac{x-y+2}{7}$
• C. $\frac{x-y-2}{7}$
• D. none of these

Answer 1

Answer: (A)

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
$\Delta =\frac{1}{2}\{x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)$
$+x_3\left(y_1-y_2\right)\}$
Given points are $A(6,3),B(-3,5),C(4,-2)$ and $P(x,y)$
$\therefore \frac{\Delta PBC}{\Delta ABC}$
$=\frac{\frac{1}{2}}{\frac{1}{2}}\left[\frac{x(5+2)-3(-2-y)+4(y-5)}{6(5+2)-3(-2-3)+4(3-5)}\right]$
$=\left[\frac{7x+7y-14}{42+15-8}\right]$
$=\frac{7x+7y-14}{49}=\frac{x+y-2}{7}$