Q1.  (A) AT WHAT POINT ON THE CURVE  x squaredy squaredminus 2 x minus 4 x plus 1 equals 0 IS THE TANGENT PARALLEL TO Y-AXIS?
 
(B)  FIND THE EQUATION OF TANGENT AT T equals straight pi over 4FOR THE  CURVE X= SIN3T , Y = COS2T .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Asked by poojaparmar065 | 10th Oct, 2014, 09:13: PM

Expert Answer:

 
left parenthesis 1 right parenthesis C o n s i d e r space t h e space g i v e n space e q u a t i o n space x squared plus y squared minus 2 x minus 4 y plus 1 equals 0 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 x comma space w e space h a v e comma 2 x plus 2 y fraction numerator d y over denominator d x end fraction minus 2 minus 4 fraction numerator d y over denominator d x end fraction equals 0... left parenthesis 1 right parenthesis I f space t h e space tan g e n t space i s space p a r a l l e l space t o space y minus a x i s comma space t h e n space s l o p e equals infinity rightwards double arrow fraction numerator d y over denominator d x end fraction equals infinity.... left parenthesis 2 right parenthesis F r o m 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 fraction numerator d y over denominator d x end fraction equals fraction numerator minus 2 x plus 2 over denominator 2 y minus 4 end fraction.... left parenthesis 3 right parenthesis 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 comma space w e space h a v e comma 2 y minus 4 equals 0 rightwards double arrow y equals 2 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 i n space t h e space g i v e n space e q u a t i o n l comma space w e space h a v e comma space x squared plus 4 minus 2 x minus 8 plus 1 equals 0 rightwards double arrow x squared minus 2 x minus 3 equals 0 rightwards double arrow open parentheses x minus 3 close parentheses open parentheses x plus 1 close parentheses equals 0 rightwards double arrow x minus 3 equals 0 space o r space x plus 1 equals 0 rightwards double arrow x equals 3 space o r space x equals minus 1 T h u s space a t space p o i n t s space left parenthesis 3 comma 2 right parenthesis space a n d space left parenthesis minus 1 comma 2 right parenthesis space t h e space tan g e n t s space t o space t h e space c u r v e space a r e space p a r a l l e l space t o space y minus a x i s
 
left parenthesis 2 right parenthesis C o n s i d e r space t h e space g i v e n h space c u r v e : x equals sin 3 t ; y equals cos 2 t fraction numerator d x over denominator d t end fraction equals 3 cos 3 t ; fraction numerator d y over denominator d t end fraction equals minus 2 sin 2 t fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d t end fraction cross times fraction numerator d t over denominator d x end fraction equals fraction numerator minus 2 sin 2 t over denominator 3 cos 3 t end fraction rightwards double arrow m equals open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript t equals straight pi over 4 end subscript equals fraction numerator minus 2 sin open parentheses 2 cross times straight pi over 4 close parentheses over denominator 3 cos open parentheses 3 cross times straight pi over 4 close parentheses end fraction equals fraction numerator minus 2 over denominator 3 cross times open parentheses begin display style fraction numerator minus 1 over denominator square root of 2 end fraction end style close parentheses end fraction equals fraction numerator 2 square root of 2 over denominator 3 end fraction... left parenthesis 1 right parenthesis T h e space p o i n t space o n space t h e space c u r v e space a t space t equals straight pi over 4 space i s comma space x subscript 1 equals sin open parentheses 3 cross times straight pi over 4 close parentheses comma space y subscript 1 equals cos open parentheses 2 cross times straight pi over 4 close parentheses rightwards double arrow x subscript 1 equals sin open parentheses fraction numerator 3 straight pi over denominator 4 end fraction close parentheses comma space y subscript 1 equals cos open parentheses straight pi over 2 close parentheses rightwards double arrow x subscript 1 equals fraction numerator 1 over denominator square root of 2 end fraction comma space y subscript 1 equals 0.... left parenthesis 2 right parenthesis  T h e space e q u a t i o n space o f space t h e space l i n e space h a v i n g space s l o p e space m space a n d space p a s sin g space t h r o u g h space t h e space p o i n t space left parenthesis x subscript 1 comma y subscript 1 right parenthesis space i s y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma rightwards double arrow y minus 0 equals fraction numerator 2 square root of 2 over denominator 3 end fraction open parentheses x minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses rightwards double arrow 3 y equals 2 square root of 2 x minus 2 T h e space e q u a t i o n space o f space t h e space tan g e n t space i s space 3 y equals 2 square root of 2 x minus 2

Answered by Vimala Ramamurthy | 11th Oct, 2014, 06:12: PM