Obtain the differential equation of the family of circles :
with centre at (1, 2)

Answer

To obtain the differential equation of the family of circles with center at (1, 2), we can follow these steps:

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

### Step 2: Differentiate the equation with respect to x
Now, we differentiate both sides of the equation with respect to x:
\[ \frac{d}{dx}[(x - 1)^2 + (y - 2)^2] = \frac{d}{dx}[r^2] \]
Since r is a constant, the derivative of \(r^2\) is 0. Applying the chain rule:
\[ 2(x - 1) + 2(y - 2)\frac{dy}{dx} = 0 \]

### Step 3: Simplify the differentiated equation
We can simplify the equation:
\[ 2(x - 1) + 2(y - 2)\frac{dy}{dx} = 0 \]
Dividing through by 2 gives:
\[ (x - 1) + (y - 2)\frac{dy}{dx} = 0 \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to isolate \(\frac{dy}{dx}\):
\[ (y - 2)\frac{dy}{dx} = - (x - 1) \]
Now, divide both sides by \(y - 2\) (assuming \(y \neq 2\)):
\[ \frac{dy}{dx} = -\frac{x - 1}{y - 2} \]

### Step 5: Final form of the differential equation
Thus, the differential equation of the family of circles with center at (1, 2) is:
\[ \frac{dy}{dx} = \frac{1 - x}{y - 2} \]