Question

The lines $a_1x + b_1y + c_1 = 0$, $a_2x + b_2y + c_2 - 0$ and $a_3x = 0$ are concurrent, if $b_1c_2 - b_2c_1$ is equal to

• A. $10$
• B. $1$
• C. $0$
• D. $-1$

Answer 1

Answer: (C)

$0$

Answer 2

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 $a_{1}x+b_{1}y+c_{1}=0\quad\ldots\left(i\right)$ $a_{2}x+b_{2}y+c_{2}=0\quad \ldots \left(ii\right)$ $a_{3}x=0\quad\ldots\left(iii\right)$ 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 $\left(ii\right)$. $\therefore \left(a_{2} \times 0\right)+b_{2}\left(\frac{-c_{1}}{b_{1}}\right)+c_{2}=0$ $\Rightarrow b_{1}c_{2}-b_{2}c_{1}=0$