A tangent drawn to the curve y = f(x) at P(x, y) cuts the x and y axes at A and B, respectively, such that AP : PB = 1 : 3. If f(1) = 1 then the curve passes through `(k,(1)/(8))` where k is

Options

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

Explanation

To solve the problem, we need to find the value of \( k \) such that the curve \( y = f(x) \) passes through the point \( (k, \frac{1}{8}) \). The tangent at point \( P(x, y) \) intersects the x-axis and y-axis at points \( A \) and \( B \) respectively, with the ratio \( AP : PB = 1 : 3 \). We know that \( f(1) = 1 \).

### Step-by-Step Solution:

1. **Identify Points**: Let the coordinates of point \( P \) be \( (x, y) \), point \( A \) be \( (\alpha, 0) \), and point \( B \) be \( (0, \beta) \).

2. **Use Section Formula**: Since \( AP : PB = 1 : 3 \), we can apply the section formula to find the coordinates of point \( P \):
\[ x = \frac{3\alpha + 0 \cdot 1}{3 + 1} = \frac{3\alpha}{4} \]
\[ y = \frac{3 \cdot 0 + 1 \cdot \beta}{3 + 1} = \frac{\beta}{4} \]

3. **Express \( \alpha \) and \( \beta \)**: From the above equations, we can express \( \alpha \) and \( \beta \) in terms of \( x \) and \( y \):
\[ \alpha = \frac{4x}{3} \]
\[ \beta = 4y \]

4. **Equation of the Tangent**: The equation of the line passing through points \( A \) and \( B \) can be written in the intercept form:
\[ \frac{x}{\alpha} + \frac{y}{\beta} = 1 \]
Substituting \( \alpha \) and \( \beta \):
\[ \frac{x}{\frac{4x}{3}} + \frac{y}{4y} = 1 \]
Simplifying this gives:
\[ \frac{3}{4} + \frac{1}{4} = 1 \]

5. **Find the Slope**: The slope of the tangent line can be derived from the equation:
\[ \text{slope} = -\frac{\beta}{\alpha} = -\frac{4y}{\frac{4x}{3}} = -\frac{3y}{x} \]

6. **Differential Equation**: The slope can also be expressed as:
\[ \frac{dy}{dx} = -\frac{3y}{x} \]

7. **Separate Variables**: Rearranging gives:
\[ \frac{dy}{y} = -3 \frac{dx}{x} \]

8. **Integrate**: Integrating both sides:
\[ \ln |y| = -3 \ln |x| + C \]
This simplifies to:
\[ \ln |y| = \ln |C| - 3 \ln |x| \implies y = \frac{C}{x^3} \]

9. **Find Constant \( C \)**: We know \( f(1) = 1 \):
\[ 1 = \frac{C}{1^3} \implies C = 1 \]
Thus, the equation of the curve is:
\[ y = \frac{1}{x^3} \]

10. **Find \( k \)**: We need to find \( k \) such that:
\[ \frac{1}{8} = \frac{1}{k^3} \]
This leads to:
\[ k^3 = 8 \implies k = 2 \]

### Final Answer:
The value of \( k \) is \( 2 \).