Point of Intersection Formula - Practice Questions & MCQ

Point of intersection of two lines

If the equations of two non-parallel lines are

$
\begin{aligned}
& L_1=a_1 x+b_1 y+c_1=0 \\
& L_2=a_2 x+b_2 y+c_2=0
\end{aligned}
$
If $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ is a point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, then solving these two equations of the line by cross multiplication

$
\frac{x_1}{b_1 c_2-c_1 b_2}=\frac{y_1}{c_1 a_2-a_1 c_2}=\frac{1}{a_1 b_2-b_1 a_2}
$
We get,

$
\left(\mathrm{x}_1, \mathrm{y}_1\right)=\left(\frac{\mathrm{b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}, \frac{\mathrm{c}_1 \mathrm{a}_2-\mathrm{c}_2 \mathrm{a}_1}{\mathrm{a}_1 \mathrm{~b}_2-\mathrm{a}_2 \mathrm{~b}_1}\right)
$
Concurrent Lines
If three straight lines meet in a point then three given lines are called concurrent.
To check if three lines are concurrent or not
1. First find the point of intersection of any two straight lines by solving them simultaneously. If this point satisfies the third equation then three lines are concurrent.

$
\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|=0
$