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 differential equation, we need to differentiate the tangent line equation with respect to \( x \):
\[ \frac{dy}{dx} = m \]
Here, \( m \) is treated as a constant.

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

### Step 5: Multiply through by \( \frac{dy}{dx} \)
To eliminate the fraction, multiply the entire equation by \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} y = \left(\frac{dy}{dx}\right)^2 x + a \]

### Step 6: Rearrange the equation
Rearranging gives us:
\[ \frac{dy}{dx} y - \left(\frac{dy}{dx}\right)^2 x - a = 0 \]

### Step 7: Identify the degree of the differential equation
The highest order of the derivative in the equation is \( \frac{dy}{dx} \). In this case, the highest power of \( \frac{dy}{dx} \) is 2 (from \( \left(\frac{dy}{dx}\right)^2 \)). Therefore, the degree of the differential equation is 2.

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

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