Question

If the straight lines $a_{1}x+b_{1}y+c_1=0$ at $a_{2}x+b_{2}y+C_2=0$ are perpendicular to each other, then

• A. $a_1{b}_1+a_2{b}_2=0$
• B. $a_1{a}_2-b_1{b}_2=0$
• C. $a_1{b}_2+a_2{b}_1=0$
• D. $a_1{a}_2+b_1{b}_2=0$

Answer 1

The correct option is D $a_1{a}_2+b_1{b}_2=0$

The equations of the lines are

$a_{1}x+b_{1}y+c_1=0$

Therefore, the slope $m_1=(-\frac{a_1}{b_1})$

and $a_{2}x+b_{2}y+c_2=0$

Therefore, the slope $m_2=(-\frac{a_2}{b_2})$

If the lines are perpendicular to each other, then

$m_1\times m_2=-1$

$\Rightarrow (-\frac{a_1}{b_1})\times (-\frac{a_2}{b_2})=-1$

$\Rightarrow a_1{a}_2+b_1{b}_2=0$