The degree of the differential equation of all tangent lines to the parabola `y^2 = 4ax` is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To find the degree of the differential equation of all tangent lines to the parabola \( y^2 = 4ax \), we can follow these steps:

### Step 1: Write the equation of the parabola
The given parabola is:
\[ y^2 = 4ax \]

### Step 2: Write the equation of the tangent line
The equation of the tangent line to the parabola can be expressed as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent line.

### Step 3: Differentiate the tangent line equation
To find the slope \( \frac{dy}{dx} \), we differentiate the equation of the tangent line:
\[ \frac{dy}{dx} = m \]

### Step 4: Substitute \( m \) in terms of \( \frac{dy}{dx} \)
From the tangent line equation, we can express \( m \) as:
\[ m = \frac{dy}{dx} \]
Substituting this into the tangent line equation gives:
\[ y = \frac{dy}{dx}x + \frac{a}{\frac{dy}{dx}} \]

### Step 5: Rearrange the equation
Rearranging the equation, we get:
\[ y - \frac{dy}{dx}x - \frac{a}{\frac{dy}{dx}} = 0 \]
Multiplying through by \( \frac{dy}{dx} \) to eliminate the fraction:
\[ y \cdot \frac{dy}{dx} - \left(\frac{dy}{dx}\right)^2 x - a = 0 \]

### Step 6: Rearranging into standard form
Rearranging gives us:
\[ \left(\frac{dy}{dx}\right)^2 x - y \cdot \frac{dy}{dx} + a = 0 \]

### Step 7: Divide by \( x \)
Dividing the entire equation by \( x \) (assuming \( x \neq 0 \)):
\[ \left(\frac{dy}{dx}\right)^2 - \frac{y}{x} \cdot \frac{dy}{dx} + \frac{a}{x} = 0 \]

### Step 8: Identify the order and degree
In this equation, the highest derivative is \( \left(\frac{dy}{dx}\right)^2 \), which indicates that the order of the differential equation is 1 (since we have only the first derivative). The degree of the differential equation is determined by the highest power of the highest derivative, which in this case is 2.

### Conclusion
Thus, the degree of the differential equation of all tangent lines to the parabola \( y^2 = 4ax \) is:
\[ \text{Degree} = 2 \]