The curve `((x)/(a))^(n)+((y)/(b))^(n)=2` touches the line `(x)/(a)+(y)/(b)=2` at the point p(a,b). The smallest prime value of n for this to be possible is :

Options

A. 2
B. 3
C. 7
D. 13

Explanation

To solve the problem, we need to determine the smallest prime 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 \( P(a, b) \).

### Step 1: Differentiate the curve equation

We start by differentiating the curve equation with respect to \( x \):

\[ \frac{d}{dx}\left(\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n\right) = \frac{d}{dx}(2) \]

Using the chain rule, we get:

\[ n\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} + n\left(\frac{y}{b}\right)^{n-1} \cdot \frac{dy}{dx} \cdot \frac{1}{b} = 0 \]

### Step 2: Rearranging the equation

Rearranging gives us:

\[ n\left(\frac{y}{b}\right)^{n-1} \cdot \frac{dy}{dx} = -n\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} \]

Dividing both sides by \( n \) (assuming \( n \neq 0 \)):

\[ \left(\frac{y}{b}\right)^{n-1} \cdot \frac{dy}{dx} = -\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} \]

### Step 3: Find \( \frac{dy}{dx} \) at the point \( P(a, b) \)

Now substituting \( x = a \) and \( y = b \):

\[ \left(\frac{b}{b}\right)^{n-1} \cdot \frac{dy}{dx} = -\left(\frac{a}{a}\right)^{n-1} \cdot \frac{1}{a} \]

This simplifies to:

\[ 1 \cdot \frac{dy}{dx} = -\frac{1}{a} \]

Thus, we have:

\[ \frac{dy}{dx} = -\frac{1}{a} \]

### Step 4: Find the slope of the line

The slope of the line \( \frac{x}{a} + \frac{y}{b} = 2 \) can be found by rewriting it in slope-intercept form:

\[ y = -\frac{b}{a}x + 2b \]

The slope of this line is \( -\frac{b}{a} \).

### Step 5: Set the slopes equal

Since the curve touches the line at the point \( P(a, b) \), the slopes must be equal:

\[ -\frac{1}{a} = -\frac{b}{a} \]

This implies:

\[ 1 = b \]

### Step 6: Substitute back to find \( n \)

Now we need to ensure that the curve touches the line. For the curve to touch the line, it must satisfy the condition at the point \( P(a, b) \):

Substituting \( b = 1 \) into the curve equation:

\[ \left(\frac{a}{a}\right)^n + \left(\frac{1}{1}\right)^n = 2 \]

This simplifies to:

\[ 1 + 1 = 2 \]

This is satisfied for any \( n \).

### Conclusion

Since we are looking for the smallest prime number \( n \), the smallest prime number is \( 2 \).

Thus, the answer is:

\[ \boxed{2} \]