find the shortest distance of the point (0,C) from the parabola y space equals space x squared where 0 less or equal than C less or equal than space 5

Asked by vinitapahel123 | 29th Nov, 2015, 05:10: PM

Expert Answer:

L e t space P open parentheses x comma y close parentheses space b e space a n y space p o i n t space o n space t h e space p a r a b o l a space y equals x squared T h e space d i s t a n c e space b e t w e e n space P open parentheses x comma y close parentheses space a n d space Q open parentheses 0 comma c close parentheses space i s space L. L equals square root of open parentheses x minus 0 close parentheses squared plus open parentheses y minus c close parentheses squared end root rightwards double arrow L equals square root of x squared plus open parentheses y minus c close parentheses squared end root rightwards double arrow L equals square root of y plus open parentheses y minus c close parentheses squared end root... left parenthesis 1 right parenthesis D i f f e r e n t i a t i n g space t h e space l e n g t h space w i t h space r e s p e c t space t o space y comma space w e space h a v e fraction numerator d L over denominator d y end fraction equals fraction numerator 1 over denominator 2 square root of open square brackets y plus open parentheses y minus c close parentheses squared close square brackets end root end fraction cross times open square brackets 1 plus 2 open parentheses y minus c close parentheses close square brackets E q u a t i n g space fraction numerator d L over denominator d y end fraction equals 0 comma space w e space h a v e comma fraction numerator 1 over denominator 2 square root of open square brackets y plus open parentheses y minus c close parentheses squared close square brackets end root end fraction cross times open square brackets 1 plus 2 open parentheses y minus c close parentheses close square brackets equals 0 rightwards double arrow 1 plus 2 open parentheses y minus c close parentheses equals 0 rightwards double arrow 2 open parentheses y minus c close parentheses equals negative 1 rightwards double arrow y minus c equals fraction numerator negative 1 over denominator 2 end fraction rightwards double arrow y equals c minus fraction numerator negative 1 over denominator 2 end fraction rightwards double arrow y equals fraction numerator 2 c minus 1 over denominator 2 end fraction... left parenthesis 2 right parenthesis A g a i n space d i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space y comma space w e space h a v e comma fraction numerator d squared L over denominator d y squared end fraction equals fraction numerator 2 square root of open square brackets y plus open parentheses y minus c close parentheses squared close square brackets end root cross times 2 minus open square brackets 1 plus 2 open parentheses y minus c close parentheses close square brackets cross times 2 cross times fraction numerator 1 over denominator 2 square root of open square brackets y plus open parentheses y minus c close parentheses squared close square brackets end root end fraction cross times open square brackets 1 plus 2 open parentheses y minus c close parentheses close square brackets over denominator 4 open square brackets y plus open parentheses y minus c close parentheses squared close square brackets end fraction W h e n space w e space s u b s t i t u t e space y equals fraction numerator 2 c minus 1 over denominator 2 end fraction comma space fraction numerator d squared L over denominator d y squared end fraction space i s space p o s i t i v e space a n d space h e n c e space i t space i s space m i n i m u m. S u b s t i t u t i n g space t h e space v a l u e space o f space y space f r o m space e q u a t i o n space left parenthesis 2 right parenthesis space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma L equals square root of fraction numerator 2 c minus 1 over denominator 2 end fraction plus open parentheses fraction numerator 2 c minus 1 over denominator 2 end fraction minus c close parentheses squared end root space space space equals square root of fraction numerator 2 c minus 1 over denominator 2 end fraction plus open parentheses fraction numerator 2 c minus 1 minus 2 c over denominator 2 end fraction close parentheses squared end root space space space equals square root of fraction numerator 2 c minus 1 over denominator 2 end fraction plus open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses squared end root space space space equals square root of fraction numerator 2 c minus 1 over denominator 2 end fraction plus 1 fourth end root space space space equals square root of fraction numerator 2 open parentheses 2 c minus 1 close parentheses over denominator 4 end fraction plus 1 fourth end root space space space equals square root of fraction numerator 4 c minus 2 plus 1 over denominator 4 end fraction end root space space space equals square root of fraction numerator 4 c minus 1 over denominator 4 end fraction end root

Answered by Vimala Ramamurthy | 30th Nov, 2015, 06:53: PM