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 system of curves given by the equation \(\left(\frac{dy}{dx}\right)^2 = \frac{a}{x}\), we will follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ \left(\frac{dy}{dx}\right)^2 = \frac{a}{x} \]
Taking the square root of both sides, we have:
\[ \frac{dy}{dx} = \pm \sqrt{\frac{a}{x}} \]

### Step 2: Find the slope of the orthogonal trajectories
The slope of the orthogonal trajectories is the negative reciprocal of the original slope. Therefore, we have:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{\frac{a}{x}}} = -\frac{\sqrt{x}}{\sqrt{a}} \]

### Step 3: Rewrite the equation for integration
We can rewrite the equation as:
\[ dy = -\frac{\sqrt{x}}{\sqrt{a}} dx \]

### Step 4: Integrate both sides
Now we will integrate both sides:
\[ \int dy = -\frac{1}{\sqrt{a}} \int \sqrt{x} \, dx \]
The left side integrates to:
\[ y = -\frac{1}{\sqrt{a}} \cdot \frac{2}{3} x^{3/2} + C \]
where \(C\) is the constant of integration.

### Step 5: Simplify the expression
Thus, we can express \(y\) as:
\[ y = -\frac{2}{3\sqrt{a}} x^{3/2} + C \]

### Step 6: Rearrange the equation
Rearranging gives us the final form of the orthogonal trajectories:
\[ 3\sqrt{a}y + 2x^{3/2} = C' \]
where \(C' = 3C\) is a new constant.

### Final Result
The orthogonal trajectories of the given system of curves are given by:
\[ 3\sqrt{a}y + 2x^{3/2} = C' \]

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