The differential equation of the family of circles with fixed radius 5 units and centre on the line
`y=2` is `(y-2)^(2){1+((dy)/(dx))^(2)}=25`
True or False?

Answer

To determine whether the statement about the differential equation of the family of circles with a fixed radius of 5 units and centers on the line \( y = 2 \) is true or false, we will derive the differential equation step by step.

### Step 1: Write the equation of a circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
For our case, the radius \(r = 5\) and the center lies on the line \(y = 2\). Therefore, we can express the center as \((h, 2)\), where \(h\) is a variable representing the x-coordinate of the center.

### Step 2: Substitute the center and radius into the circle equation
Substituting \(k = 2\) and \(r = 5\) into the circle equation, we get:
\[ (x - h)^2 + (y - 2)^2 = 5^2 \]
This simplifies to:
\[ (x - h)^2 + (y - 2)^2 = 25 \]

### Step 3: Differentiate the equation with respect to \(x\)
To eliminate the arbitrary constant \(h\), we differentiate both sides of the equation with respect to \(x\):
\[ \frac{d}{dx}[(x - h)^2 + (y - 2)^2] = \frac{d}{dx}[25] \]
Using the chain rule, we differentiate the left side:
\[ 2(x - h)(1 - \frac{dh}{dx}) + 2(y - 2)\frac{dy}{dx} = 0 \]
This simplifies to:
\[ (x - h)(1 - \frac{dh}{dx}) + (y - 2)\frac{dy}{dx} = 0 \]

### Step 4: Solve for \(\frac{dh}{dx}\)
Rearranging gives:
\[ (x - h)(1 - \frac{dh}{dx}) = -(y - 2)\frac{dy}{dx} \]
This can be solved for \(\frac{dh}{dx}\):
\[ 1 - \frac{dh}{dx} = -\frac{(y - 2)\frac{dy}{dx}}{(x - h)} \]
\[ \frac{dh}{dx} = 1 + \frac{(y - 2)\frac{dy}{dx}}{(x - h)} \]

### Step 5: Eliminate \(h\)
From the original circle equation, we can express \(h\) as:
\[ h = x \pm \sqrt{25 - (y - 2)^2} \]
We can substitute this expression for \(h\) back into the differentiated equation and simplify.

### Step 6: Final form of the differential equation
After substituting and simplifying, we arrive at:
\[ (y - 2)^2(1 + \left(\frac{dy}{dx}\right)^2) = 25 \]

### Conclusion
Thus, we have derived the differential equation:
\[ (y - 2)^2(1 + \left(\frac{dy}{dx}\right)^2) = 25 \]
This confirms that the statement is **True**.