Orthogonal trajectories of the system of curves `((dy)/(dx))^(2) = (a)/(x)` are

Options

A. `9a(y+c)^(2) = 4x^(3)`
B. `y + c = (-2)/(9sqrt(a))x^(3//2)`
C. `y^(2)+c = (2)/(3sqrt(a)) x^(3//2)`
D. `9a (y+c)^(2) = 4x^(2)`

Answer

To find the orthogonal trajectories of the given system of curves defined by the differential equation \((\frac{dy}{dx})^2 = \frac{a}{x}\), we will follow these steps:

### Step 1: Rewrite the given equation
The given equation can be rewritten as:
\[ \frac{dy}{dx} = \pm \sqrt{\frac{a}{x}} \]

### Step 2: Separate variables
We will separate the variables \(y\) and \(x\):
\[ dy = \pm \sqrt{\frac{a}{x}} \, dx \]

### Step 3: Integrate both sides
Integrating both sides, we have:
\[ \int dy = \pm \int \sqrt{\frac{a}{x}} \, dx \]
The left side integrates to \(y\), and the right side can be simplified:
\[ y = \pm \sqrt{a} \int x^{-\frac{1}{2}} \, dx \]
The integral of \(x^{-\frac{1}{2}}\) is:
\[ \int x^{-\frac{1}{2}} \, dx = 2\sqrt{x} + C \]
Thus, we have:
\[ y = \pm 2\sqrt{a}\sqrt{x} + C \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ y - C = \pm 2\sqrt{a}\sqrt{x} \]

### Step 5: Finding the orthogonal trajectories
The slopes of the orthogonal trajectories will be the negative reciprocal of the slopes of the original curves. The slope of the original curves is:
\[ \frac{dy}{dx} = \pm \sqrt{\frac{a}{x}} \]
Thus, the slope of the orthogonal trajectories is:
\[ \frac{dy}{dx} = \mp \frac{1}{\sqrt{\frac{a}{x}}} = \mp \frac{\sqrt{x}}{\sqrt{a}} \]

### Step 6: Separate variables for orthogonal trajectories
Separating variables for the orthogonal trajectories gives us:
\[ dy = \mp \frac{\sqrt{x}}{\sqrt{a}} \, dx \]

### Step 7: Integrate both sides again
Integrating both sides:
\[ \int dy = \mp \frac{1}{\sqrt{a}} \int \sqrt{x} \, dx \]
The left side integrates to \(y\), and the right side can be simplified:
\[ y = \mp \frac{1}{\sqrt{a}} \cdot \frac{2}{3} x^{\frac{3}{2}} + C \]

### Step 8: Final form of the orthogonal trajectories
Thus, the orthogonal trajectories can be expressed as:
\[ y = \mp \frac{2}{3\sqrt{a}} x^{\frac{3}{2}} + C \]

### Conclusion
The orthogonal trajectories of the system of curves defined by \((\frac{dy}{dx})^2 = \frac{a}{x}\) are given by:
\[ y = \mp \frac{2}{3\sqrt{a}} x^{\frac{3}{2}} + C \]