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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, are parallel or coincident: $6 x - 3 y + 10 = 0; 2 x - y + 9 = 0$ .

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Intersect at a point

Parallel

Coincident

Data insufficient

answer is B.

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Detailed Solution

Given linear equation are:

6 x - 3 y + 10 = 0; 2 x - y + 9 = 0

\Rightarrow 6 x - 3 y + 10 = 0

……………(1)

\Rightarrow 2 x - y + 9 = 0

……………(2)
Comparing with the general equations

a_{1} x + b_{1} y + c_{1} = 0; a_{2} x + b_{2} y + c_{2} = 0

\Rightarrow a_{1} = 6, b_{1} = - 3, c_{1} = 10, a_{2} = 2, b_{2} = - 1, c_{2} = 9

\Rightarrow \frac{a_{1}}{a_{2}} = \frac{6}{3}

\Rightarrow \frac{a_{1}}{a_{2}} = \frac{3}{1}

\Rightarrow \frac{b_{1}}{b_{2}} = \frac{- 3}{- 1}

\Rightarrow \frac{c_{1}}{c_{2}} = \frac{10}{9}

Comparing

\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}

\Rightarrow \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

Hence, the line represented by equation (1) and equation (2) are parallel.
Therefore, option (2) is correct.

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