Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin ?

Options

A. `(y - x "" (dy)/(dx))^(2) = 1 - ((dy)/(dx))^(2)`
B. `(y + x "" (dy)/(dx))^(2) = 1 + ((dy)/(dx))^(2)`
C. `(y - x "" (dy)/(dx))^(2) = 1 + ((dy)/(dx))^(2)`
D. `(y + x"" (dy)/(dx))^(2) = 1 - ((dy)/(dx))^(2)`

Answer

To solve the problem of finding the differential equation that represents the family of straight lines at a unit distance from the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation of a Line**:
The general equation of a straight line can be expressed as:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

2. **Distance from the Origin**:
The distance \( d \) from a point \((x_0, y_0)\) to a line given by \( Ax + By + C = 0 \) is calculated using the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( y = mx + c \), we can rewrite it in the standard form:
\[ mx - y + c = 0 \]
Here, \( A = m \), \( B = -1 \), and \( C = c \).

3. **Setting the Distance to 1**:
We want the distance from the origin (0, 0) to the line to be equal to 1:
\[ 1 = \frac{|c|}{\sqrt{m^2 + 1}} \]
Rearranging gives:
\[ |c| = \sqrt{m^2 + 1} \]

4. **Substituting for \( c \)**:
We can express \( c \) in terms of \( m \):
\[ c = \sqrt{m^2 + 1} \]
Thus, substituting \( c \) back into the line equation gives:
\[ y = mx + \sqrt{m^2 + 1} \]

5. **Differentiating the Equation**:
To eliminate \( m \), we differentiate the equation with respect to \( x \):
\[ \frac{dy}{dx} = m \]

6. **Substituting \( m \) Back**:
We substitute \( m = \frac{dy}{dx} \) into the line equation:
\[ y = \frac{dy}{dx} x + \sqrt{\left(\frac{dy}{dx}\right)^2 + 1} \]

7. **Rearranging the Equation**:
To eliminate the square root, we can square both sides:
\[ (y - \frac{dy}{dx} x)^2 = 1 + \left(\frac{dy}{dx}\right)^2 \]

8. **Final Form**:
Rearranging gives us the final differential equation:
\[ y - \frac{dy}{dx} x = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]

### Conclusion:
The differential equation representing the family of straight lines at a unit distance from the origin is:
\[ y - \frac{dy}{dx} x = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]