Question

The two lines $a_1 x+b_1 y+c_1=0 $ and $a_2 x+b_2 y+c_2=0$, where $b_1, b_2 \neq 0$ are
I. Parallel, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}$.
II. Perpendicular, if $a_1 a_2-b_1 b_2=0$.

• A. Both I and II are true
• B. Only I is true
• C. Only II is true
• D. Both I and II are false

Answer 1

Answer: (B)

Given lines can be written as
$y=-\frac{a_1}{b_1} x-\frac{c_1}{b_1} ....$(i)
and $ y-\frac{a_2}{b_2} x-\frac{c_2}{b_2} ....$(ii)
Slopes of the lines (i) and (ii) are $m_1=-\frac{a_1}{b_1}$ and $m_2=-\frac{a_2}{b_2}$, respectively. Now,
I. Lines are parallel, if $m_1=m_2$, which gives
$-\frac{a_1}{b_1}=-\frac{a_2}{b_2}$
or $\frac{a_1}{b_1}=\frac{a_2}{b_2}$
II. Lines are perpendicular, if $m_1, m_2=-1$, which gives
$\frac{a_1}{b_1} \cdot \frac{a_2}{b_2}=-1$
or $ a_1 a_2+b_1 b_2=0$