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Equation of a Line Perpendicular to a Given Line

A line AB mee...

Question

A line AB meets X-axis at A and Y-axis at B. P(4, -1) divides AB in the ratio 1 : 2.

(i) Find the co-ordinates of A and B.

(ii) Find the equation of the line through P and perpendicular to AB.

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Solution

$left parenthesis i right parenthesis P o i n t space A space i s space a t space X minus a x i s space colon space t h e n space A space s h o u l b space b e space left parenthesis x comma 0 right parenthesis s i m i l i a r l y space comma space P o i n t space B space i s space a t space y minus a x i s space colon space t h e n space B space s h o u l d space b e space left parenthesis 0 comma y right parenthesis B y space s e c t i o n space f o r m u l a colon fraction numerator 2 cross times x plus 1 cross times 0 over denominator 1 plus 2 end fraction equals 4 space space space space space space a n d space space fraction numerator 2 cross times 0 plus 1 cross times y over denominator 2 plus 1 end fraction equals negative 1 fraction numerator 2 x over denominator 3 end fraction equals 4 space space space space space space space space space space space space space space space space space space space space a n d space space space space space y over 3 equals negative 1 x equals 6 space space space space space a n d space space y equals negative 3 bold italic p bold italic o bold italic i bold italic n bold italic t bold space bold italic A bold space bold left parenthesis bold 6 bold comma bold 0 bold right parenthesis bold space bold space bold space bold italic a bold italic n bold italic d bold space bold italic p bold italic o bold italic i bold italic n bold italic t bold space bold italic B bold left parenthesis bold 0 bold comma bold minus bold 3 bold right parenthesis left parenthesis i i right parenthesis w e space h a v e space t o space f i n d space e q u a t i o n space o f space t h e space l i n e space t h r o u g h space P space a n d space p e r p e n d i c u l a r space t o space A B. t h e n colon s l o p e space o f space l i n e space equals fraction numerator negative 1 over denominator s l o p e space o f space A B end fraction s l o p e space o f space A B equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction equals fraction numerator 0 minus open parentheses negative 3 close parentheses over denominator 6 minus 0 end fraction equals 3 over 6 equals 1 half t h e n space s l o p e space o f space t h e space l i n e space equals fraction numerator negative 1 over denominator s l o p e space o f space A B end fraction equals fraction numerator negative 1 over denominator open parentheses 1 half close parentheses end fraction equals negative 2 h e n c e space comma space e q u a t i o n space o f space t h e space l i n e space rightwards double arrow space y minus y subscript 1 equals m left parenthesis x minus x subscript 1 right parenthesis rightwards double arrow y minus left parenthesis negative 3 right parenthesis equals negative 2 left parenthesis x minus 0 right parenthesis rightwards double arrow y plus 3 equals negative 2 x rightwards double arrow bold 2 bold italic x bold plus bold italic y bold plus bold 3 bold equals bold 0 bold space bold space bold space bold space bold space bold left parenthesis bold italic e bold italic q bold italic u bold italic a bold italic t bold italic i bold italic o bold italic n bold space bold italic o bold italic f bold space bold italic t bold italic h bold italic e bold space bold italic l bold italic i bold italic n bold italic e bold right parenthesis$

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