Question

The lines $ax+by+c=0$ and $a^{\prime }x+b^{\prime }y+c^{\prime }=0$ are

Answer 1

$L_1:ax+by+c=0$
$L_2:a^{\prime }x+b^{\prime }y+c^{\prime }=0$
A) If $L_1\&L_2$ are parallel
then $\frac{a}{a^{\prime }}=\frac{b}{b^{\prime }}$
Therefore, $a{b}^{\prime }-a^{\prime }b=0$
B) If $L_1\&L_2$ are perpendicular
then $\frac{a}{b}\frac{a^{\prime }}{b^{\prime }}=-1$
Therefore, $a{a}^{\prime }+b{b}^{\prime }=0$
C) If $L_1\&L_2$ are identical
then $\frac{a}{a^{\prime }}=\frac{b}{b^{\prime }}=\frac{c}{c^{\prime }}$
therefore, $a{b}^{\prime }{c}^{\prime }=a^{\prime }{b}^{\prime }c=a^{\prime }{c}^{\prime }b$
D) If $L_1\&L_2$ intersect $(1,1)$
then $a+b+c=a^{\prime }+b^{\prime }+c^{\prime }=0$