If a curve is such that line joining origin to any point `P (x,y)` on the curve and the line parallel to y-axis through P are equally inclined to tangent to curve at P, then the differential equation of the curve is:

Options

A. `x ((dy)/(dx))^2 -2y (dy)/(dx) =x `
B. `x ((dy)/(dx))^(2) +2y (dy)/(dx) =x `
C. `y ((dy)/(dx))^(2) -2x (dy)/(dx) =x `
D. `y ((dy)/(dx))^(2) -2y (dy)/(dx) =x `

Answer

To solve the problem, we need to derive the differential equation of the curve based on the given conditions. Let's break it down step by step.

### Step 1: Understand the Geometry
Let \( P(x, y) \) be a point on the curve. The line joining the origin \( O(0, 0) \) to the point \( P \) has a slope given by:
\[ m_{OP} = \frac{y}{x} \]
The tangent to the curve at point \( P \) has a slope \( \frac{dy}{dx} \).

### Step 2: Identify the Line Parallel to the Y-axis
The line parallel to the y-axis through point \( P \) is vertical, which means its slope is undefined or can be considered as infinite.

### Step 3: Equally Inclined Lines
Since the line \( OP \) and the vertical line through \( P \) are equally inclined to the tangent at \( P \), we can denote the angle of inclination between the tangent and the line \( OP \) as \( \theta \). The angle between the tangent and the vertical line is also \( \theta \).

### Step 4: Use the Tangent Formula
Using the tangent of the angle between two lines, we have:
\[ \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \]
For the line \( OP \) and the tangent, we have:
\[ \tan(\theta) = \frac{\frac{y}{x} - \frac{dy}{dx}}{1 + \frac{y}{x} \cdot \frac{dy}{dx}} \]
For the vertical line (which has an infinite slope) and the tangent, we have:
\[ \tan(\theta) = \frac{m_L - \frac{dy}{dx}}{1 + m_L \cdot \frac{dy}{dx}} \]
Since \( m_L \) is infinite, we can consider \( \frac{1}{m_L} = 0 \).

### Step 5: Set Up the Equation
Equating the two expressions for \( \tan(\theta) \):
\[ \frac{\frac{y}{x} - \frac{dy}{dx}}{1 + \frac{y}{x} \cdot \frac{dy}{dx}} = 0 \]
This implies:
\[ \frac{y}{x} - \frac{dy}{dx} = 0 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{y}{x} \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ x \frac{dy}{dx} - y = 0 \]

### Step 7: Form the Differential Equation
Now, we can express this as:
\[ x \frac{dy}{dx} - y = 0 \]
This is the required differential equation of the curve.

### Final Result
The differential equation of the curve is:
\[ x \frac{dy}{dx} - y = 0 \]