What is the differential equation of all circles whose centres are at the origin is

Options

A. `(dy)/(dx) = (y)/(x)`
B. `(dy)/(dx) = (x)/(y)`
C. `(dy)/(dx) =- (x)/(y)`
D. None of these

Answer

To find the differential equation of all circles whose centers are at the origin, we can follow these steps:

### Step 1: Write the equation of the circle
The general equation of a circle with center at the origin (0, 0) and radius \( r \) is given by:
\[ x^2 + y^2 = r^2 \]

### Step 2: Differentiate the equation
To find the differential equation, we differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(r^2) \]
Since \( r \) is a constant, the derivative of \( r^2 \) is 0. Thus, we have:
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 0 \]

### Step 3: Apply the differentiation
Using the chain rule, we differentiate \( y^2 \):
\[ 2x + 2y \frac{dy}{dx} = 0 \]

### Step 4: Simplify the equation
Now, we can simplify this equation:
\[ 2y \frac{dy}{dx} = -2x \]
Dividing both sides by 2 gives:
\[ y \frac{dy}{dx} = -x \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now, we can isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\frac{x}{y} \]

### Conclusion
The differential equation of all circles whose centers are at the origin is:
\[ \frac{dy}{dx} = -\frac{x}{y} \]

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