The orthogonal trajectories of the circle `x^(2)+y^(2)-ay=0`, (where a is a parameter), is

Options

A. `x^(2)+y^(2)-ay=0`
B. `x^(2)+y^(2)=Cx`
C. `x^(2)+y^(2)=C`
D. `x^(2)+y^(2)=C(x+y)`

Explanation

To find the orthogonal trajectories of the circle given by the equation \( x^2 + y^2 - ay = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle Equation
The equation of the circle can be rewritten as:
\[ x^2 + (y - \frac{a}{2})^2 = \frac{a^2}{4} \]
This shows that the circle is centered at \( (0, \frac{a}{2}) \) with a radius of \( \frac{a}{2} \).

### Step 2: Differentiate the Circle Equation
We differentiate the equation \( x^2 + y^2 - ay = 0 \) with respect to \( x \):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) - a\frac{dy}{dx} = 0 \]
This gives us:
\[ 2x + 2y\frac{dy}{dx} - a\frac{dy}{dx} = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Rearranging the equation, we have:
\[ 2y\frac{dy}{dx} - a\frac{dy}{dx} = -2x \]
Factoring out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}(2y - a) = -2x \]
Thus,
\[ \frac{dy}{dx} = \frac{-2x}{2y - a} \]

### Step 4: Find the Slope of Orthogonal Trajectories
The slope of the orthogonal trajectories is the negative reciprocal of \(\frac{dy}{dx}\):
\[ \frac{dy}{dx}_{\text{orthogonal}} = -\frac{1}{\frac{dy}{dx}} = \frac{2y - a}{2x} \]

### Step 5: Set Up the Differential Equation
We can set up the differential equation for the orthogonal trajectories:
\[ \frac{dy}{dx} = \frac{2y - a}{2x} \]

### Step 6: Separate Variables
Rearranging gives:
\[ (2y - a) dy = 2x dx \]

### Step 7: Integrate Both Sides
Integrating both sides:
\[ \int (2y - a) dy = \int 2x dx \]
This results in:
\[ y^2 - ay = x^2 + C \]
where \( C \) is the constant of integration.

### Step 8: Final Form of the Orthogonal Trajectories
Rearranging gives us the equation of the orthogonal trajectories:
\[ y^2 - ay - x^2 = C \]

### Summary of Steps:
1. Rewrite the circle equation.
2. Differentiate the equation.
3. Solve for \(\frac{dy}{dx}\).
4. Find the slope of orthogonal trajectories.
5. Set up the differential equation.
6. Separate variables.
7. Integrate both sides.
8. Rearrange to get the final form.