The equation of the curve whose subnormal is constant is

Options

A. `y=ax+b`
B. `y^(2)=2ax+b`
C. `ay^(2)-x^(2)=a`
D. none of these

Answer

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 any point is defined as the length of the perpendicular from the point on the curve to the x-axis. Mathematically, the subnormal \( S \) can be expressed as:
\[ S = y \frac{dy}{dx} \]
where \( y \) is the function of \( x \) and \( \frac{dy}{dx} \) is the derivative of \( y \) with respect to \( x \).

### Step 2: Set up the equation for a constant subnormal
According to the problem, the subnormal is constant. Let's denote this constant as \( c \):
\[ y \frac{dy}{dx} = c \]

### Step 3: Rearrange the equation
We can rearrange this equation to separate the variables:
\[ \frac{dy}{dx} = \frac{c}{y} \]

### Step 4: Integrate both sides
Now we will integrate both sides. First, we can rewrite the equation as:
\[ y \, dy = c \, dx \]
Integrating both sides gives:
\[ \int y \, dy = \int c \, dx \]
This results in:
\[ \frac{y^2}{2} = cx + C \]
where \( C \) is the constant of integration.

### Step 5: Solve for \( y^2 \)
To express \( y^2 \) in terms of \( x \), we multiply both sides by 2:
\[ y^2 = 2cx + 2C \]

### Step 6: Replace constants
Let \( 2c = a \) and \( 2C = b \) where \( a \) and \( b \) are new constants. Thus, we can rewrite the equation as:
\[ y^2 = ax + b \]

### Final Result
The equation of the curve whose subnormal is constant is:
\[ y^2 = 2ax + b \]