If $(x, y)$ be on the line joining the two points $(1, -3)$ and $(-4, 2)$, prove that $x + y + 2 = 0$.
Given:
$(x, y)$ is on the line joining the two points $(1, -3)$ and $(-4, 2)$.
To do:
We have to prove that $x + y + 2 = 0$.
Solution:
We know that,
If the points $A, B$ and $C$ are collinear then the area of $\triangle ABC$ is zero.
Let $A(1, -3), B(x, y)$ and $C(-4, 2)$ be the vertices of $\triangle ABC$.
Area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ is given by,
Area of $\Delta=\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]$
Therefore,
Area of triangle \( ABC=\frac{1}{2}[1(y-2)+x(2+3)+(-4)(-3-y)] \)
\( 0=\frac{1}{2}[y-2+x(5)-(-4)(3+y)] \)
\( 0(2)=(y-2+5x+12+4y) \)
\( 0=5x+5y+10 \)
\( 5(x+y+2)=0 \)
\( x+y+2=0 \)
Hence proved.
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