The order of the differential equation of the family of parabolas symmetric about `y=1` and tangent to x = 2 is

Options

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

Answer

To find the order of the differential equation of the family of parabolas symmetric about the line \( y = 1 \) and tangent to the line \( x = 2 \), we can follow these steps:

### Step 1: Understand the symmetry about \( y = 1 \)
Since the parabolas are symmetric about the line \( y = 1 \), we can express the general equation of the parabolas in the form:
\[ y - 1 = a(x - 2)^2 \]
where \( a \) is a parameter that defines the specific parabola in the family.

### Step 2: Identify the point of tangency
The parabolas are tangent to the line \( x = 2 \). This means that at \( x = 2 \), the value of \( y \) will be \( y = 1 \) (the vertex of the parabola) when \( a = 0 \). Thus, the vertex of the parabola lies on the line \( x = 2 \).

### Step 3: Differentiate to find the order
To find the differential equation, we need to eliminate the parameter \( a \). We start by differentiating the equation of the parabola with respect to \( x \):
\[ \frac{dy}{dx} = 2a(x - 2) \]
This gives us the first derivative.

### Step 4: Eliminate the parameter
To eliminate \( a \), we can express it in terms of \( y \) and \( x \):
From the original equation:
\[ y - 1 = a(x - 2)^2 \implies a = \frac{y - 1}{(x - 2)^2} \]
Substituting this into the derivative:
\[ \frac{dy}{dx} = 2 \left(\frac{y - 1}{(x - 2)^2}\right)(x - 2) = \frac{2(y - 1)}{(x - 2)} \]

### Step 5: Form the differential equation
Now, we can rearrange this to form a differential equation:
\[ \frac{dy}{dx} (x - 2) = 2(y - 1) \]

### Step 6: Determine the order of the differential equation
The resulting equation is a first-order differential equation since it involves only the first derivative \( \frac{dy}{dx} \) and does not involve any higher derivatives. Therefore, the order of the differential equation is 1.

### Conclusion
The order of the differential equation of the family of parabolas symmetric about \( y = 1 \) and tangent to \( x = 2 \) is:
\[ \text{Order} = 1 \]