The order and degree of the differential equation of all tangent lines to the parabola `y^(2)=4x` are

Options

A. 2,2
B. 3,1
C. 1,2
D. 4,1

Answer

To find the order and degree of the differential equation of all tangent lines to the parabola given by the equation \( y^2 = 4x \), we will follow these steps:

### Step 1: Differentiate the given equation
Start with the equation of the parabola:
\[ y^2 = 4x \]
Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \]
Using the chain rule on the left side, we get:
\[ 2y \frac{dy}{dx} = 4 \]
Thus, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \]

### Step 2: Write the equation of the tangent line
The slope of the tangent line at any point \( (x_1, y_1) \) on the parabola is given by \( \frac{2}{y_1} \). The equation of the tangent line can be expressed as:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = \frac{2}{y_1} \) and using the point \( (x_1, y_1) \) where \( y_1^2 = 4x_1 \):
\[ y - y_1 = \frac{2}{y_1}(x - x_1) \]

### Step 3: Express \( x_1 \) in terms of \( y_1 \)
From the parabola equation \( y_1^2 = 4x_1 \), we can express \( x_1 \) as:
\[ x_1 = \frac{y_1^2}{4} \]
Substituting this into the tangent line equation:
\[ y - y_1 = \frac{2}{y_1}\left(x - \frac{y_1^2}{4}\right) \]

### Step 4: Rearranging the tangent line equation
Rearranging gives:
\[ y - y_1 = \frac{2}{y_1}x - \frac{2y_1}{4} \]
This simplifies to:
\[ y = \frac{2}{y_1}x - \frac{y_1}{2} + y_1 \]
\[ y = \frac{2}{y_1}x + \frac{y_1}{2} \]

### Step 5: Eliminate the parameter \( y_1 \)
To eliminate \( y_1 \), we can express \( y_1 \) in terms of \( y \) and \( x \):
\[ y_1 = \frac{2x}{y} \implies y_1^2 = \frac{4x^2}{y^2} \]
Substituting \( y_1 \) back into the equation:
\[ y = \frac{2}{\frac{2x}{y}}x + \frac{\frac{2x}{y}}{2} \]
This leads to:
\[ y^2 = 2xy + \frac{4x^2}{y^2} \]
Multiplying through by \( y^2 \) to eliminate the fraction:
\[ y^4 = 2xy^3 + 4x^2 \]

### Step 6: Form the differential equation
Now differentiate the equation with respect to \( x \) to find the differential equation:
\[ \frac{d}{dx}(y^4) = \frac{d}{dx}(2xy^3 + 4x^2) \]
Using the product rule and chain rule, we can derive the necessary relationships.

### Conclusion: Determine the order and degree
After differentiating and simplifying, we will find that the highest derivative is \( \frac{dy}{dx} \) (first order), and the highest power of \( \frac{dy}{dx} \) in the equation will be 2. Thus, we conclude:
- **Order**: 1
- **Degree**: 2