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

Options

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

Answer

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

### Step 1: Equation of the Tangent Line
Assume the equation of the tangent line to the parabola can be written in the slope-intercept form:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Condition for Tangency
For the line to be tangent to the parabola \( x^2 = 4y \), it must intersect the parabola at exactly one point. We substitute \( y \) from the tangent line equation into the parabola equation:
\[ x^2 = 4(mx + c) \]
This simplifies to:
\[ x^2 - 4mx - 4c = 0 \]
For this quadratic equation in \( x \) to have exactly one solution (i.e., the line is tangent to the parabola), the discriminant must be zero:
\[ D = b^2 - 4ac = (−4m)^2 - 4(1)(−4c) = 16m^2 + 16c = 0 \]
Thus, we have:
\[ 16m^2 + 16c = 0 \implies c = -m^2 \]

### Step 3: Substitute \( c \) Back
Substituting \( c \) back into the equation of the tangent line gives:
\[ y = mx - m^2 \]

### Step 4: Differentiate the Tangent Line Equation
Now, we differentiate the equation \( y = mx - m^2 \) with respect to \( x \):
\[ \frac{dy}{dx} = m \]

### Step 5: Differentiate Again
Differentiating again with respect to \( x \) gives:
\[ \frac{d^2y}{dx^2} = 0 \]

### Step 6: Form the Differential Equation
From the first derivative, we have \( \frac{dy}{dx} = m \) and from the second derivative, we have \( \frac{d^2y}{dx^2} = 0 \). The first derivative \( m \) can be expressed in terms of \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = 0 \]
This is a second-order differential equation.

### Step 7: Determine Order and Degree
- The **order** of the differential equation is the highest derivative present, which is 2.
- The **degree** of the differential equation is the power of the highest derivative when the equation is a polynomial in derivatives. Here, the highest derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 1.

Thus, the order and degree of the differential equation of all tangent lines to the parabola \( x^2 = 4y \) is:
- **Order:** 2
- **Degree:** 1