Necessary and Sufficient Conditions for the General Second Degree Equation
$$ Ax^2+2Hxy+By^2+2Gx+2Fy+C=0 $$
to Represent Pair of Straight Lines:
$$
\Delta =\begin{vmatrix}
A & H &G\\
H& B & F\\
G& F& C
\end{vmatrix} =0
$$
Case 1 :if $$ H^2 > AB $$ Then they are Intersecting pair of Straight Lines
Case 2: if $$ H^2 = AB $$ Then they are pair of Parallel Straight Lines
Case 3: if $$ H^2 < AB $$ Then they Represent a Point in a Plane
$$ Ax^2+2Hxy+By^2+2Gx+2Fy+C=0 $$
to Represent Pair of Straight Lines:
$$
\Delta =\begin{vmatrix}
A & H &G\\
H& B & F\\
G& F& C
\end{vmatrix} =0
$$
Case 1 :if $$ H^2 > AB $$ Then they are Intersecting pair of Straight Lines
Case 2: if $$ H^2 = AB $$ Then they are pair of Parallel Straight Lines
Case 3: if $$ H^2 < AB $$ Then they Represent a Point in a Plane
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