If $R (x, y)$ is a point on the line segment joining the points $P (a, b)$ and $Q (b, a)$, then prove that $x + y = a + b$.
Given:
$R(x, y)$ is a point on the line segment joining the points $P(a, b)$ and $Q(b, a)$.
To do:
We have to prove that $x + y = a + b$.
Solution:
We know that,
If the points $A, B$ and $C$ are collinear then the area of $\triangle ABC$ is zero.
Let $P(a, b), R(x, y)$ and $Q(b, a)$ be the vertices of $\triangle PRQ$.
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 \( PRQ=\frac{1}{2}[a(y-a)+x(a-b)+b(b-y)] \)
\( 0=\frac{1}{2}[ay-a^2+ax-bx+b^2-by] \)
\( 0(2)=[y(a-b)+x(a-b)-(a^2-b^2)] \)
\( 0=[x(a-b)+y(a-b)-(a-b)(a+b)] \)
\( (a-b)[x+y-a-b]=0 \)
This implies,
\( a-b=0\) or \( x+y-a-b=0 \)
\( a=b \) or \( x+y=a+b \)
Hence proved.
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