Any curve which cuts every member of a given family of curve is called an orthogonal trajectory of family. To get this, we replace `(dy)/(dx) by -(dx)/(dy)` in differential equation.
The orthogonal trajectories for `y=cx^(2)`, where c is a constant is

Options

A. a family of circles
B. family of straight lines
C. family of ellipses
D. family of hyperbolas

Answer

To find the orthogonal trajectories of the family of curves given by \( y = cx^2 \), where \( c \) is a constant, we can follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the family of curves:
\[ y = cx^2 \]
Differentiating both sides with respect to \( x \):
\[ \frac{dy}{dx} = 2cx \]

### Step 2: Express \( c \) in terms of \( y \) and \( x \)
From the equation \( y = cx^2 \), we can express \( c \) as:
\[ c = \frac{y}{x^2} \]
Substituting this value of \( c \) back into the derivative:
\[ \frac{dy}{dx} = 2\left(\frac{y}{x^2}\right)x = \frac{2y}{x} \]

### Step 3: Replace \( \frac{dy}{dx} \) with \( -\frac{dx}{dy} \)
To find the orthogonal trajectories, we replace \( \frac{dy}{dx} \) with \( -\frac{dx}{dy} \):
\[ -\frac{dx}{dy} = \frac{2y}{x} \]
This implies:
\[ \frac{dx}{dy} = -\frac{2y}{x} \]

### Step 4: Separate variables and integrate
Rearranging gives:
\[ x \, dx = -2y \, dy \]
Now, we integrate both sides:
\[ \int x \, dx = \int -2y \, dy \]
This results in:
\[ \frac{x^2}{2} = -y^2 + C \]
where \( C \) is the constant of integration.

### Step 5: Rearranging the equation
Multiplying through by 2 to eliminate the fraction:
\[ x^2 + 2y^2 = 2C \]
Letting \( k = 2C \), we can rewrite the equation as:
\[ \frac{x^2}{k} + \frac{y^2}{\frac{k}{2}} = 1 \]
This represents the equation of an ellipse.

### Conclusion
Thus, the orthogonal trajectories of the family of curves \( y = cx^2 \) are given by the family of ellipses described by:
\[ \frac{x^2}{k} + \frac{y^2}{\frac{k}{2}} = 1 \]