Find the curves for which the length of normal is equal to the radius vector.

Options

A. circles
B. rectangular hyperbola
C. ellipses
D. straight lines

Explanation

To solve the problem of finding the curves for which the length of the normal is equal to the radius vector, we can follow these steps:

### Step 1: Understand the Definitions
The length of the normal at a point on a curve is given by the formula:
\[ L_n = y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
The radius vector (the distance from the origin to the point \((x, y)\)) is given by:
\[ L_r = \sqrt{x^2 + y^2} \]

### Step 2: Set Up the Equation
According to the problem, we need to set the length of the normal equal to the radius vector:
\[ y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{x^2 + y^2} \]

### Step 3: Square Both Sides
To eliminate the square roots, we square both sides of the equation:
\[ y^2 (1 + \left(\frac{dy}{dx}\right)^2) = x^2 + y^2 \]

### Step 4: Simplify the Equation
Expanding the left side gives us:
\[ y^2 + y^2 \left(\frac{dy}{dx}\right)^2 = x^2 + y^2 \]
Subtracting \(y^2\) from both sides results in:
\[ y^2 \left(\frac{dy}{dx}\right)^2 = x^2 \]

### Step 5: Rearranging the Equation
We can rearrange this to isolate \(\frac{dy}{dx}\):
\[ \left(\frac{dy}{dx}\right)^2 = \frac{x^2}{y^2} \]
Taking the square root of both sides gives us two cases:
\[ \frac{dy}{dx} = \frac{x}{y} \quad \text{or} \quad \frac{dy}{dx} = -\frac{x}{y} \]

### Step 6: Solve the Differential Equations
1. **For \(\frac{dy}{dx} = \frac{x}{y}\)**:
- This can be rewritten as:
\[ y \, dy = x \, dx \]
- Integrating both sides:
\[ \frac{y^2}{2} = \frac{x^2}{2} + C_1 \implies y^2 - x^2 = C_1 \]

2. **For \(\frac{dy}{dx} = -\frac{x}{y}\)**:
- This can be rewritten as:
\[ y \, dy = -x \, dx \]
- Integrating both sides:
\[ \frac{y^2}{2} = -\frac{x^2}{2} + C_2 \implies y^2 + x^2 = C_2 \]

### Step 7: Identify the Curves
The first equation \(y^2 - x^2 = C_1\) represents a rectangular hyperbola, while the second equation \(y^2 + x^2 = C_2\) represents a circle.

### Final Result
Thus, the curves for which the length of the normal is equal to the radius vector are:
1. Rectangular hyperbolas: \(y^2 - x^2 = C_1\)
2. Circles: \(x^2 + y^2 = C_2\)

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