Understanding the Concept
In coordinate geometry, a homogeneous second-degree equation of the form $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin. However, when we have a general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, it can represent a pair of straight lines (if the determinant condition is met).
Regardless of the linear terms ($gx, fy, c$), the angle between the lines depends only on the coefficients of the second-degree terms ($x^2, xy, y^2$). Specifically, the angle $\theta$ between the lines represented by $ax^2 + 2hxy + by^2 + \dots = 0$ is given by the formula:
$\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$
Solving the Problem
Given Equation: $x^2 + 6xy + 9y^2 + 4x + 12y - 5 = 0$
Step 1: Identify coefficients
Comparing the given equation with the standard form $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$:
- $a = 1$
- $2h = 6 \implies h = 3$
- $b = 9$
Step 2: Apply the angle formula
Substitute the values into the formula $\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$:
$\tan \theta = \left| \frac{2\sqrt{3^2 - (1)(9)}}{1 + 9} \right|$
Step 3: Simplify
$\tan \theta = \left| \frac{2\sqrt{9 - 9}}{10} \right|$ $\tan \theta = \left| \frac{2\sqrt{0}}{10} \right|$ $\tan \theta = 0$
Step 4: Conclusion
Since $\tan \theta = 0$, it implies that $\theta = 0^\circ$ (or $180^\circ$).
Intuition
When the discriminant $h^2 - ab = 0$, the pair of lines is either parallel or coincident. In our case, notice that the expression $x^2 + 6xy + 9y^2$ is a perfect square: $(x + 3y)^2$. This indicates that the lines are actually parallel to each other. Because the lines are parallel, the angle between them is $0^\circ$.