The degree of the differential equation of all curves having normal of constant length 'c' is

Options

A. 1
B. 3
C. 4
D. none of these

Answer

To find the degree of the differential equation of all curves having a normal of constant length 'c', we can follow these steps:

### Step 1: Understanding the Length of the Normal
The length of the normal to a curve at a point \((x, y)\) is given by the formula:
\[ L = y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
where \(L\) is the length of the normal.

### Step 2: Setting Up the Equation
According to the problem, the length of the normal is a constant \(c\). Therefore, we can write:
\[ y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = c \]

### Step 3: Squaring Both Sides
To eliminate the square root, we square both sides of the equation:
\[ \left(y \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\right)^2 = c^2 \]
This simplifies to:
\[ y^2 \left(1 + \left(\frac{dy}{dx}\right)^2\right) = c^2 \]

### Step 4: Expanding the Equation
Expanding the left-hand side gives us:
\[ y^2 + y^2 \left(\frac{dy}{dx}\right)^2 = c^2 \]

### Step 5: Rearranging the Equation
Rearranging the equation, we have:
\[ y^2 \left(\frac{dy}{dx}\right)^2 = c^2 - y^2 \]

### Step 6: Identifying the Degree
Now, we need to identify the degree of the differential equation. The highest order derivative in this equation is \(\frac{dy}{dx}\), which is of order 1. The power of this derivative in the equation is 2 (since it appears squared).

Thus, the degree of the differential equation is:
\[ \text{Degree} = 2 \]

### Conclusion
The degree of the differential equation of all curves having a normal of constant length \(c\) is 2.