Question

The lines $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2-0$ and $a_{3}x=0$ are concurrent, if $b_1{c}_2-b_2{c}_1$ is equal to

• A. $10$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

Answer: (C)

Three lines are said to be concurrent, if they pass through a common point, i.e., point of intersection of any two lines lies on the third line.
We have
$a_{1}x+b_{1}y+c_1=0\dots \left(i\right)$
$a_{2}x+b_{2}y+c_2=0\dots \left(ii\right)$
$a_{3}x=0\dots \left(iii\right)$
Let the above three lines are concurrent.
Solving $\left(i\right)$ and $\left(iii\right)$,
we get $x=0$, $y=\frac{-c}{b_1}$
$\therefore$ The point of intersection of two lines is $\left(0,\frac{-c_1}{b_1}\right)$.
Since above lines are concurrent, the point $\left(0,\frac{-c_1}{b_1}\right)$ lies on $\left(ii\right)$.
$\therefore \left(a_2\times 0\right)+b_2\left(\frac{-c_1}{b_1}\right)+c_2=0$
$\Rightarrow b_1{c}_2-b_2{c}_1=0$