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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