If point R(x,y) lies 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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The line segment P ( a,b ) and Q ( b,a ) and in it lies R ( x,y ) .
Hence the line is collinear .
The triangle formed between the 3 points will have an area = 0 .
1/2 | x₁ ( y₂ - y₃ ) + x₂ ( y₃ - y₁ ) + x₃ ( y₁ - y₂ ) | = 0
= | a ( a - y ) + b ( y - b ) + x ( b - a ) | = 0
= a² - ay + by - b² + bx - ax = 0
= bx + by - ay - ax + a² - b² = 0
= b ( x + y ) - a ( x + y ) + ( a + b )( a - b ) = 0
= ( x + y )( b - a ) - ( b - a )( a + b ) = 0
= ( b - a )( x + y - ( a + b ) ) = 0
= ( x + y ) = ( a + b ) .
Hence proved !
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