Question

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.

Answer 1

If 3 points are collinear, then the area of the triangle formed by them is zero.

Area of triangle = $\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|=0$

$\Rightarrow x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)=0$

$\Rightarrow x(b-a)+a(a-y)+b(y-b)=0$

$\Rightarrow x(b-a)+y(b-a)+a^2-b^2=0$

$\Rightarrow x(b-a)+y(b-a)-(b^2-a^2)=0$

$\Rightarrow x(b-a)+y(b-a)-(b+a)(b-a)=0$

$\Rightarrow (b-a)[x+y-(a+b\left)\right]=0$

$\Rightarrow x+y=a+b$