If the distance of a point `(x_(1),y_(1))` from each of the two straight lines, which pass through the origin of coordinates, is `delta`, then the two lines are given by

Options

A. `(x_(1)y-xy_(1))^(2)=delta^(2)(x^(2)+y^(2))`
B. `(x_(1)y+xu_(1))^(2)=delta^(2)(x^(2)+y^(2))`
C. `(x_(1)y-xy_(1))^(2)=delta^(2)(x^(2)-y^(2))`
D. none of these

Explanation

To find the two lines that pass through the origin and are at a distance of \(\delta\) from the point \((x_1, y_1)\), we can follow these steps:

### Step-by-Step Solution:

1. **Equation of the Line**:
The lines passing through the origin can be expressed in the form \(y = mx\), where \(m\) is the slope of the line. Therefore, the equation of the line can be rewritten as:
\[ mx - y = 0 \]

2. **Distance Formula**:
The distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For our line \(mx - y = 0\), we have \(A = m\), \(B = -1\), and \(C = 0\). Thus, the distance from the point \((x_1, y_1)\) to the line is:
\[ \delta = \frac{|mx_1 - y_1|}{\sqrt{m^2 + 1}} \]

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ \delta^2 = \frac{(mx_1 - y_1)^2}{m^2 + 1} \]

4. **Cross Multiplication**:
Cross-multiplying gives:
\[ \delta^2(m^2 + 1) = (mx_1 - y_1)^2 \]

5. **Expanding the Right Side**:
Expanding the right side:
\[ \delta^2 m^2 + \delta^2 = m^2x_1^2 - 2mx_1y_1 + y_1^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ m^2(x_1^2 - \delta^2) - 2mx_1y_1 + (y_1^2 - \delta^2) = 0 \]

7. **Quadratic in m**:
This is a quadratic equation in \(m\):
\[ m^2(x_1^2 - \delta^2) - 2mx_1y_1 + (y_1^2 - \delta^2) = 0 \]

8. **Using the Quadratic Formula**:
The solutions for \(m\) can be found using the quadratic formula:
\[ m = \frac{-(-2x_1y_1) \pm \sqrt{(-2x_1y_1)^2 - 4(x_1^2 - \delta^2)(y_1^2 - \delta^2)}}{2(x_1^2 - \delta^2)} \]

9. **Final Result**:
The two lines are given by the slopes \(m_1\) and \(m_2\) obtained from the quadratic equation above.