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Q. If x=acos^3¤sin^2¤ , y=asin^3¤cos^2¤ and (x^2+y^2)^p/(xy)^q (p,q belongs to N) is independent of ¤,then find value of p and q.




  • Q. If x=acos^3¤sin^2¤ , y=asin^3¤cos^2¤ and (x^2+y^2)^p/(xy)^q (p,q belongs to N) is independent of ¤,then find value of p and q.




     




Grade:11

1 Answers

Amit Saxena
35 Points
8 years ago

x^2 = a^2 cos ^6 x * sin^4 x
y^2 = a^2 sin ^6 x * cos ^4 x

adding them we get a^2 sin ^ 4 x * cos ^ 4 x   applying trigonometric identities
xy = a ^ 2 cos ^ 5x sin ^ 5 x

now putting it in the expression

we get
(sin 2x)^(4p - 5q)    ignoring the constants and applying trigonometric identities 

now the expression has to be independent of x
so put 4p - 5q = 0
so p/q = 5/4

now p , q are natural numbers
p = 5t , q = 4 t
where t belongs to positive integers

will be the set of values of p,q

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