Question

On comparing the ratios$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, or parallel or coincident:

$\begin{array}{rcl}9x+3y+12& =& 0\\ 18x+6y+24& =& 0\end{array}$

Answer 1

Step 1: Analysing the condition by which lines are intersecting, parallel or coincident :

If we are given a set of two linear equations,

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

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

• lines will intersect at a point if $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
• lines will be parallel to each other if $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
• lines will be coincident if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Step 2: Verify the given set of the equations:

The equations are -

$\begin{array}{rcl}9x+3y+12& =& 0\\ 18x+6y+24& =& 0\end{array}$

By comparing the given equations with

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

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

$a_1=9,a_2=18,b_1=3,b_2=6,c_1=12,c_2=24$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{9}{18}=\frac{1}{2}, \\ \frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2} \\ \frac{c_1}{c_2}=\frac{12}{24}=\frac{1}{2} \end{gathered}$$

Since, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, therefore the given set of equations are coincident.