The differential equation of all circles whose radius is 5 centre is any point (h,k) is

Options

A. `25((d^2y)/(dx^2))+(1+((dy)/(dx))^2)^2 =0`
B. `25((d^2y)/(dx^2))-(1+((dy)/(dx))^2)^2 =0`
C. `25((d^2y)/(dx^2))+(1+((dy)/(dx))^2)^3 =0`
D. `25((d^2y)/(dx^2))-(1+((dy)/(dx))^2)^3 =0`

Answer

To find the differential equation of all circles with a radius of 5 and center at any point (h, k), we can follow these steps:

### Step 1: Write the equation of a circle
The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Given that the radius \( r = 5 \), we can substitute this into the equation:
\[ (x - h)^2 + (y - k)^2 = 5^2 \]
This simplifies to:
\[ (x - h)^2 + (y - k)^2 = 25 \]

### Step 2: Differentiate the equation with respect to x
Now, we differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}[(x - h)^2] + \frac{d}{dx}[(y - k)^2] = \frac{d}{dx}[25] \]
Using the chain rule:
\[ 2(x - h) + 2(y - k) \frac{dy}{dx} = 0 \]
Dividing through by 2 gives:
\[ (x - h) + (y - k) \frac{dy}{dx} = 0 \]

### Step 3: Rearranging the equation
Rearranging the above equation for \( (y - k) \):
\[ (y - k) \frac{dy}{dx} = - (x - h) \]
This can be expressed as:
\[ y - k = -\frac{(x - h)}{\frac{dy}{dx}} \]

### Step 4: Differentiate again
Next, we differentiate the equation again with respect to \( x \):
\[ \frac{d}{dx}(y - k) = \frac{d}{dx}\left(-\frac{(x - h)}{\frac{dy}{dx}}\right) \]
This gives us:
\[ \frac{dy}{dx} = -\left(\frac{d}{dx}(x - h) \cdot \frac{1}{\frac{dy}{dx}} + (x - h) \cdot \frac{d}{dx}\left(\frac{1}{\frac{dy}{dx}}\right)\right) \]
Since \( \frac{d}{dx}(x - h) = 1 \), we can simplify this further.

### Step 5: Substitute and simplify
After differentiating, we can substitute back to find the relationship between \( y \), \( k \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \). We will end up with a second-order differential equation.

### Final Step: Formulate the differential equation
After performing the above steps, we can express the final differential equation in terms of \( y \), \( k \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \).

The required differential equation can be summarized as:
\[ (y - k) \frac{d^2y}{dx^2} + (1 + \left(\frac{dy}{dx}\right)^2) = 0 \]