The order of differential equation of all circles of given radius 'a' is

Options

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

Explanation

To find the order of the differential equation of all circles of a given radius 'a', we can follow these steps:

### Step 1: Write the equation of a circle
The general equation of a circle with center at point (h, k) and radius 'a' is given by:
\[ (x - h)^2 + (y - k)^2 = a^2 \]

### Step 2: Identify the variables
In this equation, we have two variables: \(x\) and \(y\). The parameters \(h\) and \(k\) represent the center of the circle, which can vary.

### Step 3: Differentiate the equation
To eliminate the parameters \(h\) and \(k\), we can differentiate the equation with respect to \(x\). This will help us find a relationship involving \(y\) and its derivatives.

Differentiating both sides gives:
\[ 2(x - h) \frac{dx}{dx} + 2(y - k) \frac{dy}{dx} = 0 \]
This simplifies to:
\[ (x - h) + (y - k) \frac{dy}{dx} = 0 \]

### Step 4: Solve for \(h\) and \(k\)
From the differentiated equation, we can express \(h\) and \(k\) in terms of \(x\), \(y\), and \(\frac{dy}{dx}\):
\[ h = x + (y - k) \frac{dy}{dx} \]
However, we need to eliminate \(h\) and \(k\) completely.

### Step 5: Differentiate again
To eliminate \(k\), we can differentiate the equation again. This will give us a second derivative:
\[ \frac{d^2y}{dx^2} \]
This leads us to a second equation involving \(y\), \(\frac{dy}{dx}\), and \(\frac{d^2y}{dx^2}\).

### Step 6: Determine the order of the differential equation
The highest derivative that appears after differentiating twice is \(\frac{d^2y}{dx^2}\). Since we have differentiated twice, the order of the differential equation is 2.

### Conclusion
Thus, the order of the differential equation of all circles of a given radius 'a' is:
\[ \text{Order} = 2 \]

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