If x cos a + y sin a = p touches the cuve x^m .y^n = a^(m+n) ;prove that

p^(m+n) . m . n = (m+n)^(m+n) cos^m a sin^n a

Simplify algebra by making subs X=x/a, Y=y/b and m=n/(n-1)

Xᵐ + Yᵐ = 1 … (i)

Xacotα + Yb = pcosecα … (ii)

Diff (i) wrt X : mXᵐ⁻¹ + mYᵐ⁻¹(dY/dX) = 0 ⟹ dY/dX = −(X/Y)ᵐ⁻¹

At pt of contact (X,Y) have grad (i) = grad (ii) so −acotα/b=−(X/Y)ᵐ⁻¹ … (iii)

Sub in (ii) for acotα : bXᵐ/Yᵐ⁻¹ + Yb = pcosecα ⟹ b(Xᵐ+Yᵐ) = pYᵐ⁻¹cosecα ⟹ Yᵐ⁻¹ = (b/p)sinα

Also using (iii), Xᵐ⁻¹ = (a/p)cosα

Sub in (i) for X,Y and adjust to get (acosα/p)^(m/(m-1)) + (bsinα/p)^(m/(m-1)) = 1

Setting m/(m-1)=n and multiply by pⁿ gives desired result