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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
The formula for the area ‘A’ encompassed by three points ( x1 , y1) , (x2 , y2) and (x3 , y3) is given by the formula,
\[∆ = \frac{1}{2}\left| \left( x_1 y_2 + x_2 y_3 + x_3 y_1 \right) - \left( x_2 y_1 + x_3 y_2 + x_1 y_3 \right) \right|\]
If three points are collinear the area encompassed by them is equal to 0.
It is said that the point R(x, y) lies on the line segment joining the points P(a, b) and Q(b, a). Hence we understand that these three points are collinear. So the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( ay + xa + b^2 \right) - \left( xb + by + a^2 \right) \right|\]
\[ 0 = \frac{1}{2}\left| ay + xa + b^2 - xb - by - a^2 \right|\]
\[ 0 = ay + xa + b^2 - xb - by - a^2 \]
\[ a^2 - b^2 = ax + ay - bx - by\]
\[ \left( a + b \right)\left( a - b \right) = \left( a - b \right)\left( x + y \right)\]
\[ \left( a + b \right) = \left( x + y \right)\]
Hence under the given conditions we have proved that x + y = a + b .
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