why are the lines perpendicular if the coefficient of x^2 +coefficient of  y^2 = 0 in a general second degree equation

In a general second-degree equation of the form:

\[
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\]

The condition for two lines to be perpendicular is determined by the relationship between the coefficients of the quadratic terms and the mixed term \(Bxy\). When two lines are perpendicular, their slopes are negative reciprocals of each other.

The general form of a conic section can represent different shapes depending on the values of \(A\), \(B\), and \(C\). In particular, for the equation to represent a pair of straight lines, it should factor into two linear factors. The lines will be perpendicular if the product of their slopes is -1.

For this to happen, the condition involving the coefficients of the quadratic terms and the mixed term \(Bxy\) is:

\[
A + C = 0
\]

This equation suggests that the sum of the coefficients of \(x^2\) and \(y^2\) must be zero.

### Why does \(A + C = 0\) imply perpendicular lines?

1. **Equation of Lines**: For the quadratic equation to represent two lines, the equation can be rewritten as:
   \[
   (L_1)(L_2) = 0
   \]
   where \(L_1\) and \(L_2\) are linear equations representing the two lines.

2. **Slope Condition**: If the lines are perpendicular, their slopes \(m_1\) and \(m_2\) must satisfy \(m_1 \cdot m_2 = -1\).

3. **Resulting Condition**: Mathematically, this condition translates to \(A + C = 0\), indicating that the lines are perpendicular. When you substitute the equation into the condition for the lines to be perpendicular, you end up with the sum of the coefficients of \(x^2\) and \(y^2\) being zero.

Therefore, when the sum of the coefficients of \(x^2\) and \(y^2\) is zero, the quadratic equation represents two perpendicular lines.

Would you like more details or have any questions?

### 5 Relative Questions:
1. How do we derive the general equation for conic sections?
2. What is the geometrical interpretation of the mixed term \(Bxy\)?
3. Can this method be applied to determine parallel lines?
4. How can we distinguish between different conic sections using the discriminant?
5. What happens if the condition \(A + C = 0\) is not satisfied?

### Tip:
To determine the nature of conic sections represented by a general quadratic equation, use the discriminant \(\Delta = B^2 - 4AC\), which helps in identifying whether the conic is an ellipse, parabola, or hyperbola.

Explore why lines are perpendicular in conic sections with a detailed explanation of the conditions in quadratic equations. Learn about the general form of a conic section and the specific condition \(A + C = 0\) for lines to be perpendicular. Suitable for advanced high school students and college-level learners.

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