For the equation of the curve whose subnormal is constant then,

Options

A. its eccentricity is 1
B. its eccentricity is `sqrt(2)`
C. its axis is the x-axis
D. its axis is the y-axis

Explanation

To find the equation of the curve whose subnormal is constant, we can follow these steps:

### Step 1: Understand the concept of subnormal
The subnormal of a curve at a point is defined as the length of the projection of the segment of the normal line between the x-axis and the curve onto the x-axis. For a function \( y = f(x) \), the subnormal \( S \) can be expressed as:

\[ S = y \cdot \frac{dy}{dx} \]

### Step 2: Set the subnormal to a constant
Since the problem states that the subnormal is constant, we can set:

\[ y \cdot \frac{dy}{dx} = k \]

where \( k \) is a constant.

### Step 3: Rearrange the equation
Rearranging the equation gives us:

\[ y \, dy = k \, dx \]

### Step 4: Integrate both sides
Now, we will integrate both sides:

\[ \int y \, dy = \int k \, dx \]

This results in:

\[ \frac{y^2}{2} = kx + C \]

where \( C \) is the constant of integration.

### Step 5: Multiply through by 2
To simplify, we can multiply the entire equation by 2:

\[ y^2 = 2kx + 2C \]

Let \( 2C \) be a new constant \( C' \):

\[ y^2 = 2kx + C' \]

### Step 6: Identify the type of curve
The equation \( y^2 = 2kx + C' \) represents a parabola.

### Step 7: Determine the orientation of the parabola
Since the equation is in the form \( y^2 = 2kx + C' \), it indicates that the parabola opens either to the right or left, depending on the sign of \( k \).

### Conclusion
Thus, the equation of the curve whose subnormal is constant is a parabola that opens along the x-axis.