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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.

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Solution

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

Area of triangle = 12|x1(y2y3)+x2(y3y1)+x3(y1y2)|=0

x1(y2y3)+x2(y3y1)+x3(y1y2)=0

x(ba)+a(ay)+b(yb)=0

x(ba)+y(ba)+a2b2=0

x(ba)+y(ba)(b2a2)=0

x(ba)+y(ba)(b+a)(ba)=0

(ba)[x+y(a+b)]=0

x+y=a+b


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