If the line `xcosalpha+ysinalpha=` touches the curve
`x^(m)y^(n)=a^(m+n)`, prove that
`p^(m+n).m^(m).n^(n)=a^(m+n)(m+n)^(m+n).cos^(m)alpha.sin^(n)alpha`

If the line x cos alpha+y sin alpha= p touches the curve x^my^n= a^(m+n) prove that p^(m+n).m^m.n^n= a^(m+n)(m+n)^(m+n).cos^m alpha. Sin^n alpha .

If the straigjht line x cos alpha+ y sin alpha=p touches the curve x^(m)y^(n)=a^(m+n) , prove that. p^(m+n)""m^(m)n^(n)=(m+n)^(m+n)a^(m+n)sin^(n) alpha cos^(m) alpha

If x^(m)*y^(n)=(x+y)^(m+n) , show that (dy)/(dx)=(y)/(x) .

If x = m cosec alpha and y = n cot alpha , then the value of (x^2)/(m^2) - (y^2)/(n^2) is

m^(3)-n^(3)-m(m^(2)-n^(2))+n(m-n)^(2)

Prove the quotient rule using a^m/a^n=a^(m-n)

Factorise : m^(4)+m^(2)n^(2)+n^(4)

If "cosec" theta- sin theta=m^(3) and sec theta-cos theta=n^(3) , then prove that m^(2)n^(2)(m^(2)+n^(2))=1.

If a^m a^n =a^(mn) , then express m in terms of n.

Simplify : (2^n.6^(m+1).10^(m-n).15^(m+n-2))/(4^m.3^(m+n).25^(m-1)) .