Question

A necessary and sufficient condition that the general equation of second degree \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) may represent a pair of straight lines is

• A. \( abc + 2fgh - af^2 - bg^2 - ch^2>0 \)
• B. \( abc + 2fgh - af^2 - bg^2 - ch^2<0 \)
• C. \( abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \)
• D. \( abc + 2fgh - af^2 - bg^2 - ch^2 = a^2 + b^2 + c^2 \)

Hint:

A useful mnemonic to remember the expression for the determinant is "A Hot Girl Had Beautiful Face, Go For Chocolates":
\(a(bc - f^2)\)
\(-h(hc - fg)\)
\(+g(hf - bg)\)
Another way is to remember the phrase: "All handsome guys having beautiful faces go for coffee", which helps to set up the matrix \(\begin{pmatrix} a & h & g
h & b & f
g & f & c \end{pmatrix}\).

Answer 1

The Correct Option is C

Step 1: Conceptualization:
A general second-degree equation defines a conic section. Under a specific coefficient constraint, this conic section degenerates into two straight lines. This constraint is determined by the determinant of a corresponding matrix.
Step 2: Methodology:
Represent the general second-degree equation in matrix form using homogeneous coordinates. The condition for the equation to represent two straight lines is that the determinant of the associated 3x3 symmetric matrix equals zero.
The matrix is:\[ \Delta = \begin{vmatrix} a & h & g
h & b & f
g & f & c \end{vmatrix} \]The condition is \( \Delta = 0 \).
Step 3: Elaboration:
Compute the determinant of the matrix \( \Delta \).\[ \det(\Delta) = a(bc - f^2) - h(hc - fg) + g(hf - bg) \]\[ = abc - af^2 - h^2c + fgh + fgh - bg^2 \]\[ = abc + 2fgh - af^2 - bg^2 - ch^2 \]For the equation to represent two straight lines, this determinant must be zero.\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]Step 4: Conclusion:
The condition for the general second-degree equation to represent a pair of straight lines is \( abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \).