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Section Formula

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Question

Calculate the ratio in which the line joining the points (-3,-1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

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Solution

The equation of line joining (x1,y1) and (x2,y2) is $fraction numerator bold italic y bold minus bold italic y subscript bold 1 over denominator bold italic y subscript bold 2 bold minus bold italic y subscript bold 1 end fraction equals fraction numerator bold italic x bold minus bold italic x subscript bold 1 over denominator bold italic x subscript bold 2 bold minus bold italic x subscript bold 1 end fraction$
$H e n c e space e q u a t i o n space o f space l i n e space j o i n i n g space left parenthesis negative 3 comma negative 1 right parenthesis space a n d space left parenthesis 5 comma 7 right parenthesis space i s fraction numerator y minus left parenthesis negative 1 right parenthesis over denominator 7 minus left parenthesis negative 1 right parenthesis end fraction equals fraction numerator x minus left parenthesis negative 3 right parenthesis over denominator 5 minus left parenthesis negative 3 right parenthesis end fraction rightwards double arrow fraction numerator y plus 1 over denominator 8 end fraction equals fraction numerator x plus 3 over denominator 8 end fraction rightwards double arrow y equals x plus 2 S o l u t i o n space o f space e q u a t i o n s space x equals 2 space a n d space y equals x plus 2 space g i v e s space p o i n t space o f space i n t e r s e c t i o n. P u t t i n g space x equals 2 space i n space e q u a t i o n space i n space y equals x plus 2 comma W e space g e t space t h e space p o i n t s comma space x equals 2 space a n d space y equals 4 T h e space c o minus o r d i n a t e s space o f space t h e space p o i n t space o f space i n t e r s e c t i o n space i s space left parenthesis 2 comma 4 right parenthesis. W e space u s e space s e c t i o n space f o r m u l a space t o space f i n d space t h e space r a t i o space i n space w h i c h space t h e space l i n e space d i v i d e s space i t. open parentheses fraction numerator m subscript 1 x subscript 2 plus m subscript 2 x subscript 1 over denominator m subscript 1 plus m subscript 2 end fraction comma fraction numerator m subscript 1 y subscript 2 plus m subscript 2 y subscript 1 over denominator m subscript 1 plus m subscript 2 end fraction close parentheses equals open parentheses 2 comma 4 close parentheses space open parentheses fraction numerator m subscript 1 cross times 5 plus m subscript 2 cross times left parenthesis negative 3 right parenthesis over denominator m subscript 1 plus m subscript 2 end fraction close parentheses equals 2 rightwards double arrow 5 m subscript 1 minus 3 m subscript 2 equals 2 left parenthesis m subscript 1 plus m subscript 2 right parenthesis rightwards double arrow 5 m subscript 1 minus 3 m subscript 2 equals 2 m subscript 1 plus 2 m subscript 2 rightwards double arrow 3 m subscript 1 equals 5 m subscript 2 rightwards double arrow m subscript 1 over m subscript 2 equals 5 over 3 S o space t h e space r a t i o space i n space w h i c h space t h e space l i n e space x equals 2 space d i v i d e s space t h e space t w o space p o i n t s space left parenthesis negative 3 comma negative 1 right parenthesis space a n d space left parenthesis 5 comma 7 right parenthesis i s comma m subscript 1 colon m subscript 2 equals 5 colon 3$

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