This is from R.M.Khan's Analytical Geometry:
Why did the author consider using the determinant in such particular way?
And what is the product this is referring to? It is not the usual matrix multiplication.
Can someone name the concepts used by the writer?
The author has considered the product :
$\begin{vmatrix} 2l_{ 1 }l_{ 2 } & { l_{ 1 }m_{ 2 }+l_{ 2 }m_{ 1 } } & l_{ 1 }n_{ 2 }+l_{ 2 }n_{ 1 } \\ l_{ 1 }m_{ 2 }+l_{ 2 }m_{ 1 } & 2m_{ 1 }m_{ 2 } & m_{ 1 }n_{ 2 }+m_{ 2 }n_{ 1 } \\ l_{ 1 }n_{ 2 }+l_{ 2 }n_{ 1 } & m_{ 1 }n_{ 2 }+m_{ 2 }n_{ 1 } & 2n_{ 1 }n_{ 2 } \end{vmatrix}$ (Let us name it $D$)
$= \begin{vmatrix} l_ 1 & l_2 & 0 \\ m_1 & m_2 & 0 \\ n_1 & n_2 & 0 \end{vmatrix}$ $(D_1)$ $\times$ $\begin{vmatrix} l_{ 2} & l_{ 1 } & 1 \\ m_{ 2 } & m_{ 1 } & 1 \\ n_{ 2 } & n_{ 1 } & 1 \end{vmatrix}$ $(D_2)$
Just in order to prove the condition required.
Here $0$ is used in first determinant to make the product equal to required condition determinant.No other concept.Just mathematical adjustment.
If some variables $a,b,c,f,g,h$
Satisfy the condition :
$l_1l_2=a , l_1m_2+l_2m_1=2h , m_1m_2=b , l_1n_2+l_2n_1=2g, m_1n_2+m_2n_1=2f,n_1n_2=c$
for two st.lines $l_1x+m_1y+n_1$ and $l_2x+m_2y+n_2$.
Then the equation :
$ax^2+2hxy+by^2+2gx+2fy+c=0$
must represent a pair of st. lines.
Or viceversa, If the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of st. line than $a,b,c,f,g,h$ must satisfy
$\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}=0$
Beacuse, since they represent pair of st. lines , they must satisfy
$l_1l_2=a , l_1m_2+l_2m_1=2h , m_1m_2=b , l_1n_2+l_2n_1=2g, m_1n_2+m_2n_1=2f,n_1n_2=c$
for two st.lines $l_1x+m_1y+n_1$ and $l_2x+m_2y+n_2$.
and the determinant $D$ can be written as $D_1 \times D_2$
Where $|D_1|=0$ . Hence $|D|$ also must be $=0$
P.S. : Someone might think that first determinant is multiplied to transpose of second determinant.No.There are several ways to multiply two determinants.Namely :
Row by Row
Column by Row
Row by Column
Column by Column
The result is basically the same in all.
Since, in matrix multiplication, we can use only Row by Column procedure, and many people use the same in determinant multiplication too this multiplication of determinants seems to be weird.So, there is nothing wrong or weird here as, may be, the author of this book has practice to multiply determinants Row by Row.