The equation of the straight lines joining the orign to the points of intersection of the line `x-y=2` and the curve `5x^(2)+12xy-8y^(2)+8x-4y+12=0` are `y=+-2x`. Are these lines equally inclined to the axes?

Answer

To solve the problem, we need to determine whether the lines \( y = 2x \) and \( y = -2x \) are equally inclined to the axes.

### Step-by-Step Solution:

1. **Identify the Lines**: The given lines are \( y = 2x \) and \( y = -2x \).

2. **Find the Slopes**:
- The slope of the line \( y = 2x \) is \( m_1 = 2 \).
- The slope of the line \( y = -2x \) is \( m_2 = -2 \).

3. **Calculate the Angles with the Positive X-axis**:
- For the line \( y = 2x \):
\[ \theta_1 = \tan^{-1}(m_1) = \tan^{-1}(2) \]
- For the line \( y = -2x \):
\[ \theta_2 = \tan^{-1}(m_2) = \tan^{-1}(-2) \]

4. **Determine the Relationship Between the Angles**:
- The angle \( \theta_2 \) can be expressed as:
\[ \theta_2 = \tan^{-1}(-2) = \pi + \tan^{-1}(2) \quad (\text{since } \tan^{-1}(-x) = \pi + \tan^{-1}(x) \text{ for } x > 0) \]
- Thus, we can say:
\[ \theta_2 = \pi - \theta_1 \]

5. **Check for Equally Inclined Lines**:
- Two lines are said to be equally inclined to the axes if the angles they make with the positive x-axis are equal in magnitude.
- Here, the angles \( \theta_1 \) and \( \theta_2 \) are \( \tan^{-1}(2) \) and \( \pi - \tan^{-1}(2) \) respectively.
- The absolute values of these angles are equal, which confirms that the lines are equally inclined to the axes.

### Conclusion:
Yes, the lines \( y = 2x \) and \( y = -2x \) are equally inclined to the axes.