Question

If $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is rep. POSL then.

Answer 1

The condition for the equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ to represent a pair of straight lines is: $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$

Answer 2

The general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if and only if the determinant of the associated matrix is zero.

The associated matrix is given by: $A = \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix}$

The condition for the equation to represent a pair of straight lines is $\det(A) = 0$.

Calculating the determinant: $\det(A) = a \begin{vmatrix} b & f \\ f & c \end{vmatrix} - h \begin{vmatrix} h & f \\ g & c \end{vmatrix} + g \begin{vmatrix} h & b \\ g & f \end{vmatrix}$ $\det(A) = a(bc - f^2) - h(hc - gf) + g(hf - gb)$ $\det(A) = abc - af^2 - h^2c + hgf + ghf - g^2b$ $\det(A) = abc - af^2 - ch^2 + 2fgh - bg^2$

Setting the determinant to zero gives the required condition: $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$