The differential equation of the family of circles of fixed radius `r` and having their centres on `y`-axis is:

Options

A. `y^(2)(1+y_(1)^(2))=r^(2)y_(1)^(2)`
B. `y^(2)=r^(2)y_(1)+y_(1)^(2)`
C. `x^(2)(1+y_(1)^(2))=r^(2)y_(1)^(2)`
D. `x^(2)=r^(2)y_(1)+y_(1)^(2)`

Answer

To find the differential equation of the family of circles of fixed radius \( r \) with centers on the \( y \)-axis, we can follow these steps:

### Step 1: Write the equation of the circle
The center of the circle lies on the \( y \)-axis, which we can denote as \( (0, \alpha) \). The equation of a circle with center \( (0, \alpha) \) and radius \( r \) is given by:

\[ x^2 + (y - \alpha)^2 = r^2 \]

### Step 2: Differentiate the equation
Next, we differentiate the equation of the circle with respect to \( x \):

\[ \frac{d}{dx}(x^2) + \frac{d}{dx}((y - \alpha)^2) = \frac{d}{dx}(r^2) \]

This gives us:

\[ 2x + 2(y - \alpha) \frac{dy}{dx} = 0 \]

### Step 3: Solve for \( y - \alpha \)
Rearranging the differentiated equation, we can isolate \( y - \alpha \):

\[ 2(y - \alpha) \frac{dy}{dx} = -2x \]

Dividing both sides by 2:

\[ (y - \alpha) \frac{dy}{dx} = -x \]

Thus, we have:

\[ y - \alpha = -\frac{x}{\frac{dy}{dx}} \]

### Step 4: Substitute back into the original equation
Now we substitute \( y - \alpha \) back into the original circle equation:

\[ x^2 + \left(-\frac{x}{\frac{dy}{dx}}\right)^2 = r^2 \]

This simplifies to:

\[ x^2 + \frac{x^2}{\left(\frac{dy}{dx}\right)^2} = r^2 \]

### Step 5: Multiply through by \( \left(\frac{dy}{dx}\right)^2 \)
To eliminate the fraction, we multiply through by \( \left(\frac{dy}{dx}\right)^2 \):

\[ x^2 \left(\frac{dy}{dx}\right)^2 + x^2 = r^2 \left(\frac{dy}{dx}\right)^2 \]

### Step 6: Rearranging the equation
Rearranging gives us:

\[ x^2 \left(\frac{dy}{dx}\right)^2 - r^2 \left(\frac{dy}{dx}\right)^2 + x^2 = 0 \]

Factoring out \( \left(\frac{dy}{dx}\right)^2 \):

\[ \left(\frac{dy}{dx}\right)^2 (x^2 - r^2) + x^2 = 0 \]

### Step 7: Final form of the differential equation
This leads us to the final form of the differential equation:

\[ x^2 \left(\frac{dy}{dx}\right)^2 + (x^2 - r^2) = 0 \]

This is the required differential equation of the family of circles of fixed radius \( r \) with centers on the \( y \)-axis.

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