Find the angle(s) between the curves y=x squared and x=y squared. Sir this is a 6 mark question . plz give the complete answer with all the steps.

Asked by bidhanrout65 | 5th Dec, 2014, 07:58: AM

Expert Answer:

G i v e n space c u r v e s space a r e space x equals y squared space a n d space y equals x squared L e t space u s space f i n d 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 o f space t h e s e space c u r v e s. rightwards double arrow x equals y squared equals open parentheses x squared close parentheses squared rightwards double arrow x equals x to the power of 4 rightwards double arrow x cubed equals 1 rightwards double arrow x equals 1 rightwards double arrow y equals 1 T h u s space t h e space i n t e r s e c t i o n space p o i n t space i s space left parenthesis 1 comma 1 right parenthesis. C o n s i d e r space t h e space g i v e n space f u n c t i o n s : x equals y squared space a n d space y equals x squared N o w space d i f f e r e n t i a t i n g space b o t h space t h e space f u n c t i o n s space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma 1 equals 2 y fraction numerator d y over denominator d x end fraction space a n d space fraction numerator d y over denominator d x end fraction equals 2 x L e t space theta subscript 1 space a n d space theta subscript 2 space b e space t h e space a n g l e s space o f space i n c l i n a t i o n space o f space t h e space tan g e n t s space o f space t h e space c u r v e s space x equals y squared space a n d space y equals x squared. T h u s comma space m subscript 1 equals tan space theta subscript 1 space a n d space m subscript 2 equals tan theta subscript 2 space space a r e space t h e space s l o p e s space o f space t h e space tan g e n t s space o f space t h e space c u r v e s space x equals y squared space a n d space y equals x squared. T h u s comma space w e space h a v e comma 1 equals 2 y m subscript 1 space a n d space m subscript 2 equals 2 x rightwards double arrow m subscript 1 equals fraction numerator 1 over denominator 2 y end fraction space a n d space m subscript 2 space equals 2 x rightwards double arrow open parentheses m subscript 1 close parentheses subscript left parenthesis 1 comma 1 right parenthesis end subscript equals fraction numerator 1 over denominator 2 cross times 1 end fraction space a n d space open parentheses m subscript 2 close parentheses subscript open parentheses 1 comma 1 close parentheses end subscript space equals 2 cross times 1 rightwards double arrow open parentheses m subscript 1 close parentheses subscript left parenthesis 1 comma 1 right parenthesis end subscript equals 1 half space a n d space open parentheses m subscript 2 close parentheses subscript open parentheses 1 comma 1 close parentheses end subscript space equals 2 L e t space theta space b e space t h e space a n g l e space b e t w e e n space t h e space c u r v e s space x equals y squared space a n d space y equals x squared. T h e n space tan theta space i s space g i v e n space b y : space tan theta equals open vertical bar fraction numerator m subscript 1 minus m subscript 2 over denominator 1 plus m subscript 1 cross times m subscript 2 end fraction close vertical bar rightwards double arrow space tan theta equals open vertical bar fraction numerator 1 half minus 2 over denominator 1 plus 1 half cross times 2 end fraction close vertical bar rightwards double arrow space tan theta equals open vertical bar fraction numerator fraction numerator minus 3 over denominator 2 end fraction over denominator 1 plus 1 end fraction close vertical bar rightwards double arrow space tan theta equals open vertical bar fraction numerator minus 3 over denominator 2 cross times 2 end fraction close vertical bar rightwards double arrow space tan theta equals open vertical bar fraction numerator minus 3 over denominator 4 end fraction close vertical bar rightwards double arrow space tan theta equals 3 over 4 rightwards double arrow space theta equals tan to the power of minus 1 end exponent open parentheses 3 over 4 close parentheses

Answered by Vimala Ramamurthy | 5th Dec, 2014, 10:27: AM