Understanding General Second-Degree Equations
A general equation of the second degree in two variables $x$ and $y$ is given by the expression:
$$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$
To determine if such an equation represents a pair of straight lines, we check a specific condition involving its coefficients. This condition is that the determinant of the symmetric matrix formed by these coefficients must be zero. Mathematically, this is expressed as:
$$\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$$
If $\Delta = 0$, the equation represents a pair of straight lines. If $\Delta \neq 0$, it does not.
Step-by-Step Solution
The given equation is:
$$x^2 + 6xy + 9y^2 + 4x + 12y - 5 = 0$$
Step 1: Identify the Coefficients
By comparing the given equation with the standard form $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, we identify the coefficients:
- $a = 1$
- $2h = 6 \implies h = 3$
- $b = 9$
- $2g = 4 \implies g = 2$
- $2f = 12 \implies f = 6$
- $c = -5$
Step 2: Calculate the Determinant $\Delta$
Now, we plug these values into the determinant formula:
$$\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$$
Substitute the values:
- $abc = (1)(9)(-5) = -45$
- $2fgh = 2(6)(3)(2) = 72$
- $-af^2 = -(1)(6)^2 = -36$
- $-bg^2 = -(9)(2)^2 = -36$
- $-ch^2 = -(-5)(3)^2 = 5(9) = 45$
Step 3: Compute the Final Sum
$$\Delta = -45 + 72 - 36 - 36 + 45$$
Combine the terms: $$\Delta = (-45 + 45) + 72 - 72$$ $$\Delta = 0 + 0 = 0$$
Conclusion
Since the determinant $\Delta = 0$, the given second-degree equation does represent a pair of straight lines.