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Question: The curve \(\left( \frac{x}{a} \right)^{n} + \left( \frac{y}{b} \right)^{n}\)= 2 touches the line \(...

The curve (xa)n+(yb)n\left( \frac{x}{a} \right)^{n} + \left( \frac{y}{b} \right)^{n}= 2 touches the line xa+yb=2\frac{x}{a} + \frac{y}{b} = 2 at the point (a, b) for n =

A

1

B

2

C

4

D

n ¹ 0

Answer

n ¹ 0

Explanation

Solution

(dydx)(a,b)\left( \frac{dy}{dx} \right)_{(a,b)}= – ((n/an)×xn−1(n/bn).yn−1)(a,b)=−ba\left( \frac{(n/a^{n}) \times x^{n - 1}}{(n/b^{n}).y^{n - 1}} \right)_{(a,b)} = - \frac{b}{a}

Tangent y – b = −ba- \frac{b}{a} (x – a)

xa+yb=2\frac{x}{a} + \frac{y}{b} = 2

Since it is independent of n, so it touches the curve for all n

for n = 0, we have no curve

\ n ¹ 0