Q1. If P(x,y) is a point on the line segment joining the points :
1) A(6,-8) and B(-8,6), prove that  x+y=2.
2) A(a,b) and B(b,a) =, then prove that x+y = a+b. 
 

Asked by tanishqa.rathod | 26th Dec, 2015, 09:58: PM

Expert Answer:

open parentheses 1 close parentheses G i v e n space t h a t space A open parentheses 6 comma negative 8 close parentheses space a n d space B open parentheses negative 8 comma 6 close parentheses space a r e space t w o space g i v e n space p o i n t s. L e t space P open parentheses x comma y close parentheses space b e space a space p o i n t space t h a t space d i v i d e s space t h e space l i n e space s e g m e n t space j o i n i n g space t h e space t w o space p o i n t s space A space a n d space B space i n space t h e space r a t i o space m colon n. H e n c e space b y space s e c t i o n space f o r m u l a comma space w e space h a v e comma x equals fraction numerator m cross times x subscript 2 plus n cross times x subscript 1 over denominator m plus n end fraction a n d y equals fraction numerator m cross times y subscript 2 plus n cross times y subscript 1 over denominator m plus n end fraction H e r e comma x equals fraction numerator negative 8 m plus 6 n over denominator m plus n end fraction... open parentheses 1 close parentheses a n d y equals fraction numerator 6 m minus 8 n over denominator m plus n end fraction... open parentheses 2 close parentheses A d d i n g space e q u a t i o n s space open parentheses 1 close parentheses space a n d space open parentheses 2 close parentheses comma space w e space h a v e comma x plus y equals fraction numerator negative 8 m plus 6 n over denominator m plus n end fraction plus fraction numerator 6 m minus 8 n over denominator m plus n end fraction space space space space space space space equals fraction numerator negative 8 m plus 6 n plus 6 m minus 8 n over denominator m plus n end fraction space space space space space space space equals fraction numerator negative 2 m minus 2 n over denominator m plus n end fraction space space space space space space space equals fraction numerator negative 2 open parentheses m plus n close parentheses over denominator m plus n end fraction space space space space space space space equals negative 2  open parentheses 2 close parentheses G i v e n space t h a t space A open parentheses a comma b close parentheses space a n d space B open parentheses b comma a close parentheses space a r e space t w o space g i v e n space p o i n t s. L e t space P open parentheses x comma y close parentheses space b e space a space p o i n t space t h a t space d i v i d e s space t h e space l i n e space s e g m e n t space j o i n i n g space t h e space t w o space p o i n t s space A space a n d space B space i n space t h e space r a t i o space m colon n. H e n c e space b y space s e c t i o n space f o r m u l a comma space w e space h a v e comma x equals fraction numerator m cross times x subscript 2 plus n cross times x subscript 1 over denominator m plus n end fraction a n d y equals fraction numerator m cross times y subscript 2 plus n cross times y subscript 1 over denominator m plus n end fraction H e r e comma x equals fraction numerator m b plus n a over denominator m plus n end fraction... open parentheses 1 close parentheses a n d y equals fraction numerator n a plus m b over denominator m plus n end fraction... open parentheses 2 close parentheses A d d i n g space e q u a t i o n s space open parentheses 1 close parentheses space a n d space open parentheses 2 close parentheses comma space w e space h a v e comma x plus y equals fraction numerator m b plus n a over denominator m plus n end fraction plus fraction numerator n a plus m b over denominator m plus n end fraction space space space space space space space equals fraction numerator m b plus n a plus n b plus m a over denominator m plus n end fraction space space space space space space space equals fraction numerator m open parentheses a plus b close parentheses plus n open parentheses a plus b close parentheses over denominator m plus n end fraction space space space space space space space equals fraction numerator open parentheses a plus b close parentheses open parentheses m plus n close parentheses over denominator m plus n end fraction space space space space space space space equals a plus b

Answered by Vimala Ramamurthy | 28th Dec, 2015, 09:05: AM