Question

The two lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$, where $b_1,b_2\ne 0$ 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 $a_1{a}_2+b_1{b}_2=0$