If OT and ON are respectively the lengths of the perpendiculars drawn from the origin to the tangent and normal drawn at any arbitrary point on the curve, x equals a cos cubed t comma space y equals a sin cubed t comma space t h e n space s h o w space t h a t space 4 O T squared plus O N squared equals a squared.

Asked by Aswin | 5th Jan, 2016, 05:43: PM

Expert Answer:

x equals a cos cubed t comma space y equals a sin cubed t fraction numerator d y over denominator d x end fraction equals fraction numerator d y divided by d t over denominator d x divided by d t end fraction equals fraction numerator 3 a s i n squared t open parentheses c o s t close parentheses over denominator 3 a c o s squared t open parentheses negative s i n t close parentheses end fraction equals negative tan t H e n c e comma space s l o p e space o f space tan g e n t space equals negative tan t S l o p e space o f space n o r m a l equals c o t t E q u a t i o n space o f space tan g e n t space a t space p o i n t space apostrophe t apostrophe space i s y minus a s i n cubed t equals negative t a n t open parentheses x minus a c o s cubed t close parentheses L e n g t h space o f space p e r p e n d i c u l a r space f r o m space o r i g i n space left parenthesis 0 comma space 0 right parenthesis equals open vertical bar fraction numerator negative a s i n cubed t minus a tan t c o s cubed t over denominator square root of 1 plus tan squared t end root end fraction close vertical bar equals open vertical bar fraction numerator a sin t open parentheses sin squared t plus cos squared t close parentheses over denominator s e c t end fraction close vertical bar equals vertical line a s i n t c o s t vertical line therefore 4 O T squared equals 4 a squared sin squared t cos squared t...... left parenthesis 1 right parenthesis E q u a t i o n space o f space n o r m a l space a t space p o i n t space apostrophe t apostrophe space i s y minus a s i n cubed t equals c o t t open parentheses x minus a c o s cubed t close parentheses L e n g t h space o f space p e r p e n d i c u l a r space f r o m space o r i g i n space left parenthesis 0 comma space 0 right parenthesis equals open vertical bar fraction numerator negative a s i n cubed t plus a c o t t c o s cubed t over denominator square root of 1 plus c o t squared t end root end fraction close vertical bar equals open vertical bar fraction numerator negative a sin to the power of 4 t plus a cos to the power of 4 t over denominator sin t cos e c t end fraction close vertical bar equals open vertical bar a open parentheses c o s to the power of 4 t minus sin to the power of 4 t close parentheses close vertical bar equals open vertical bar a open parentheses cos squared t minus sin squared t close parentheses open parentheses cos squared t plus sin squared t close parentheses close vertical bar equals open vertical bar a open parentheses c o s squared t minus s i n squared t close parentheses close vertical bar therefore O N squared equals a squared open parentheses c o s squared t minus s i n squared t close parentheses squared....... left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space 4 O T squared plus O N squared equals a squared open parentheses 4 s i n squared t c o s squared t plus open parentheses c o s squared t minus s i n squared t close parentheses squared close parentheses equals a squared open parentheses open parentheses c o s squared t plus s i n squared t close parentheses squared close parentheses equals a squared

Answered by satyajit samal | 6th Jan, 2016, 05:52: PM