Question

On comparing the ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, or are parallel or coincident.
$9x+3y+12=0;18x+6y+24=0$

• A. Intersect at a point
• B. Parallel
• C. Coincident
• D. Cannot be determined

Answer 1

The correct option is C Coincident

The given linear equation are
$\Rightarrow 9x+3y+12=\mathrm{0....}\left(1\right)$
$\Rightarrow a_1=9,b_1=3,c_1=12$
$\Rightarrow 18x+6y+24=\mathrm{0...}\left(2\right)$
$\Rightarrow a_2=18,b_2=6,c_2=24$
$\Rightarrow \frac{a_1}{a_2}=\frac{9}{18}=\frac{1}{2}$
$\Rightarrow \frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$
$\Rightarrow \frac{c_1}{c_2}=\frac{12}{24}=\frac{1}{2}$
comparing
$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$
We have $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Hence, the lines represented by eq1 and eq2 are coincident