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

Options

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

Answer

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

### Step 1: Write the equation of the tangent line
The general equation of a tangent line to the parabola can be expressed in the slope-intercept form:
\[ y = mx + c \]
where \( m \) is the slope of the tangent and \( c \) is the y-intercept.

### Step 2: Substitute into the parabola equation
For the line to be tangent to the parabola \( x^2 = 2y \), we substitute \( y \) from the tangent line equation into the parabola's equation:
\[ x^2 = 2(mx + c) \]
This simplifies to:
\[ x^2 - 2mx - 2c = 0 \]

### Step 3: Use the condition for tangency
For the line to be tangent to the parabola, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero. The discriminant \( D \) of the quadratic \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
In our case:
- \( a = 1 \)
- \( b = -2m \)
- \( c = -2c \)

Thus, the discriminant becomes:
\[ D = (-2m)^2 - 4(1)(-2c) = 4m^2 + 8c \]
Setting the discriminant to zero for tangency:
\[ 4m^2 + 8c = 0 \]
This leads to:
\[ c = -\frac{m^2}{2} \]

### Step 4: Write the equation of the tangent line
Substituting \( c \) back into the tangent line equation gives:
\[ y = mx - \frac{m^2}{2} \]

### Step 5: Differentiate the tangent equation
To find the differential equation, we differentiate the tangent line equation with respect to \( x \):
\[ \frac{dy}{dx} = m \]

### Step 6: Differentiate again
Differentiating again gives:
\[ \frac{d^2y}{dx^2} = 0 \]

### Step 7: Form the differential equation
The first derivative \( \frac{dy}{dx} = m \) can be expressed in terms of \( y \) and \( x \) as:
\[ \frac{dy}{dx} = m \quad \text{and} \quad \frac{d^2y}{dx^2} = 0 \]
This indicates that the order of the differential equation is 2 (since we have a second derivative).

### Step 8: Determine the degree
The degree of a differential equation is the power of the highest derivative when the equation is a polynomial in derivatives. Here, the highest derivative is \( \frac{d^2y}{dx^2} \) and it appears to the first power. Therefore, the degree is 1.

### Final Answer
Thus, the order of the differential equation of all tangent lines to the parabola \( x^2 = 2y \) is **2** and the degree is **1**.

---