The curve `(x/a)^n+(y/b)^n` =2 touches the straight line `x/a+y/b=2` at the point (a,b) then the value of n is

Options

A. 2
B. 3
C. 4
D. any real number

Answer

To solve the problem, we need to determine the value of \( n \) such that the curve \( \left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2 \) touches the line \( \frac{x}{a} + \frac{y}{b} = 2 \) at the point \( (a, b) \).

### Step 1: Find the derivative of the curve
We start with the equation of the curve:
\[ \left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2 \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}\left(\left(\frac{x}{a}\right)^n\right) + \frac{d}{dx}\left(\left(\frac{y}{b}\right)^n\right) = 0 \]
Using the chain rule, we get:
\[ n \left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} + n \left(\frac{y}{b}\right)^{n-1} \cdot \frac{1}{b} \cdot \frac{dy}{dx} = 0 \]

### Step 2: Rearranging the derivative
Rearranging the equation gives:
\[ n \left(\frac{y}{b}\right)^{n-1} \cdot \frac{1}{b} \cdot \frac{dy}{dx} = -n \left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} \]
Dividing both sides by \( n \) (assuming \( n \neq 0 \)):
\[ \left(\frac{y}{b}\right)^{n-1} \cdot \frac{1}{b} \cdot \frac{dy}{dx} = -\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} \]
Thus, we have:
\[ \frac{dy}{dx} = -\frac{b}{a} \cdot \frac{\left(\frac{x}{a}\right)^{n-1}}{\left(\frac{y}{b}\right)^{n-1}} \]

### Step 3: Evaluate the slope at the point (a, b)
Substituting \( x = a \) and \( y = b \):
\[ \frac{dy}{dx}\bigg|_{(a,b)} = -\frac{b}{a} \cdot \frac{\left(\frac{a}{a}\right)^{n-1}}{\left(\frac{b}{b}\right)^{n-1}} = -\frac{b}{a} \]

### Step 4: Find the slope of the line
The line \( \frac{x}{a} + \frac{y}{b} = 2 \) can be rewritten as:
\[ y = -\frac{b}{a}x + 2b \]
The slope of this line is also \( -\frac{b}{a} \).

### Step 5: Conclusion about the value of n
Since the slopes of the curve and the line are equal at the point \( (a, b) \), and since the slope does not depend on \( n \), we conclude that the value of \( n \) can be any real number.

Thus, the final answer is:
\[ \text{The value of } n \text{ is any real number.} \]