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Q:Find the point on the curve y squared=4x which is nearest to (2,-8).

Asked by Ravi consul 15th March 2015, 8:02 PM
Answered by Expert
Answer:
C o n s i d e r space t h e space e q u a t i o n space o f space t h e space c u r v e y squared equals 4 x L e t space P left parenthesis x comma y right parenthesis space b e space a space p o i n t space o n space t h e space c u r v e space s u c h space t h a t space P space i s space n e a r e s t space t o space t h e space p o i n t space left parenthesis 2 comma negative 8 right parenthesis T h u s comma space t h e space d i s tan 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 left parenthesis 2 comma negative 8 right parenthesis space i s D equals square root of open parentheses x minus 2 close parentheses squared plus open parentheses y plus 8 close parentheses squared end root S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma D squared equals open parentheses x minus 2 close parentheses squared plus open parentheses y plus 8 close parentheses squared... left parenthesis 1 right parenthesis A l s o space w e space h a v e comma space y squared equals 4 x... left parenthesis 2 right parenthesis space h e n c e comma space y to the power of 4 equals 16 x squared rightwards double arrow x squared equals y to the power of 4 over 16... left parenthesis 3 right parenthesis C o n s i d e r space t h e space e q u a t i o n space left parenthesis 1 right parenthesis colon D squared equals x squared plus 4 minus 4 x plus y squared plus 64 plus 16 y rightwards double arrow D squared equals y to the power of 4 over 16 plus 4 minus 4 x plus 4 x plus 64 plus 16 y space space space space space left square bracket f r o m space e q u a t i o n s space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis right square bracket rightwards double arrow D squared equals y to the power of 4 over 16 plus 70 plus 16 y D i f f e r e n t i a t i n g space t h e space a b o v e space e q u a t i o n 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 2 D cross times fraction numerator d D over denominator d y end fraction equals fraction numerator 4 y cubed over denominator 16 end fraction plus 16 S i n c e space w e space n e e d space t o space f i n d space t h e space m i n i m u m space o f space D comma space w e space s h o u l d space h a v e comma space fraction numerator d D over denominator d y end fraction equals 0 rightwards double arrow fraction numerator 4 y cubed over denominator 16 end fraction plus 16 equals 0 rightwards double arrow y cubed over 4 plus 16 equals 0 rightwards double arrow y cubed over 4 equals negative 16 rightwards double arrow y cubed equals negative 64 rightwards double arrow y equals negative 4 W e space h a v e comma space y squared equals 4 x rightwards double arrow 16 equals 4 x rightwards double arrow x equals 4 T h u s space t h e space p o i n t space n e a r e s t space t o space left parenthesis 2 comma negative 8 right parenthesis space i s space P open parentheses 4 comma negative 4 close parentheses
Answered by Expert 16th March 2015, 11:37 AM
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