Order of the differential equation of the family of all concentric circles centred at `(h,k)` is

Options

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

Answer

To find the order of the differential equation of the family of all concentric circles centered at the point \((h, k)\), we can follow these steps:

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

### Step 2: Identify the arbitrary constant
In this equation, \(r\) is the arbitrary constant that represents the radius of the circle. Since we are dealing with a family of concentric circles, \(r\) can take any positive value.

### Step 3: Differentiate the equation
To find the differential equation, we need to eliminate the arbitrary constant \(r\). We can differentiate the equation with respect to \(x\):
\[ \frac{d}{dx}[(x - h)^2 + (y - k)^2] = \frac{d}{dx}[r^2] \]
Using the chain rule, we get:
\[ 2(x - h) + 2(y - k)\frac{dy}{dx} = 0 \]

### Step 4: Simplify the differentiated equation
We can simplify this equation:
\[ (x - h) + (y - k)\frac{dy}{dx} = 0 \]
This is a first-order differential equation since we have only one derivative \(\frac{dy}{dx}\).

### Step 5: Determine the order of the differential equation
The order of a differential equation is determined by the highest derivative present. In our case, the highest derivative is \(\frac{dy}{dx}\), which is of order 1. Therefore, the order of the differential equation representing the family of all concentric circles is:
\[ \text{Order} = 1 \]

### Final Answer
The order of the differential equation of the family of all concentric circles centered at \((h, k)\) is **1**.

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