Question

Prove that the condition that x cos α + y sin α = p touches  the curve x^m yⁿ = a^m+n is p^Amⁿn^m = A^A ∙ a^A cos^m α ∙ sin then A =_____ p^m+n ∙m^m ∙ nⁿ =

(a) m + n 

(b) m – n 

(c) n – m 

(d) m² – n²

Answer 1

The correct option (a) m + n   

Explanation:

Carve: x^m yⁿ = a^m+n ⇒ m ∙ log x + n ∙ log y = (m + n) log a

∴ (m/x) + n ∙ (1/y) (dy/dx) = (m + n) (0)

∴ (m/x) =[(– n)/y] (dy/dx)

∴ (dy / dx) = [(– my)/(nx)]

Let point (x₁, y₁) be on curve, then equation of tangent at (x₁, y₁) is

(y – y₁) = (dy/dx)_{at [(x)1,(y)1]} (x – x₁)

i.e. (y – y₁) = [(– my₁)/(nx₁)] (x – x₁) ⇒ (mx/x₁) + (ny / y₁) = m + n (1)

also condition given is x cos α + y sin α = p (2)

∴ from (1) & (2), [(cos α)/(m/x₁)] [(sin α)/(n/y₁) = [p/(m + n)]

⇒ [(x₁ cos α)/m] = [(y₁ sin α)/n] = [(p/(m + n)

∴ x₁ = [mp/{(m + n)cos x}] and

y₁ = [np/{(m + n) sin x}]

∴ x₁^m ∙ y₁ⁿ = [{m^m p^m ∙ nⁿ ∙ pⁿ}/{(m + n)^m+n cos^m α ∙ sinⁿ α}] and x^m yⁿ = a^m+n hence

a^m+n (m + n)^m+n cos^m α ∙ sinⁿ α = m^m ∙ nⁿ ∙ p^(m+n)

but given condition is

p^A m^m nⁿ = A^A ∙ a^A ∙ cos^mα sinⁿα

by comparison, A = m + n