At the point `P(a,a^(n))` on the graph of `y=x^(n)(n in N)` in the first quadrant at normal is drawn. The normal intersects the Y-axis at the point (0, b). If `underset(ararr0)(lim)b=(1)/(2)`, then n equals ……………

Options

A. 1
B. 3
C. 2
D. 4

Answer

To solve the problem step by step, we will follow the given information and derive the necessary equations.

### Step 1: Find the derivative of the function
The function given is \( y = x^n \). To find the slope of the tangent line at the point \( P(a, a^n) \), we need to differentiate \( y \) with respect to \( x \).

\[ \frac{dy}{dx} = n x^{n-1} \]

### Step 2: Evaluate the derivative at point \( P(a, a^n) \)
Now, we substitute \( x = a \) into the derivative to find the slope at point \( P \):

\[ \frac{dy}{dx} \bigg|_{x=a} = n a^{n-1} \]

### Step 3: Find the slope of the normal line
The slope of the normal line is the negative reciprocal of the slope of the tangent line:

\[ \text{slope of normal} = -\frac{1}{n a^{n-1}} \]

### Step 4: Write the equation of the normal line
Using the point-slope form of the equation of a line, the equation of the normal line at point \( P(a, a^n) \) is:

\[ y - a^n = -\frac{1}{n a^{n-1}}(x - a) \]

Rearranging this, we get:

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

### Step 5: Find the y-intercept
To find the y-intercept (where \( x = 0 \)), we substitute \( x = 0 \) into the equation of the normal line:

\[ b = a^n + \frac{a}{n a^{n-1}} = a^n + \frac{1}{n} a^{1 - (n-1)} = a^n + \frac{1}{n} a^{2-n} \]

### Step 6: Evaluate the limit as \( a \to 0 \)
We need to evaluate the limit \( \lim_{a \to 0} b \):

\[ \lim_{a \to 0} b = \lim_{a \to 0} \left( a^n + \frac{1}{n} a^{2-n} \right) \]

### Step 7: Analyze the limit
- If \( n > 2 \), then \( a^{2-n} \to \infty \) as \( a \to 0 \).
- If \( n = 2 \), then \( b \to \frac{1}{2} \).
- If \( n < 2 \), then \( a^n \to 0 \) and \( a^{2-n} \to 0 \), thus \( b \to 0 \).

Given that \( \lim_{a \to 0} b = \frac{1}{2} \), we conclude that \( n = 2 \).

### Final Answer
Thus, the value of \( n \) is:

\[ \boxed{2} \]