In linear equation $a_{1} x + b_{1} y + c_{1} = 0$ , $a_{2} x + b_{2} y + c_{2} = 0$ the conditions for infinitely many solutions is:

\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}

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\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}

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

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

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Solution

The correct option is C $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
In case of infinite many solutions ratio of the coefficients of the equations must be same.
Thus for the equations too have infinite solutions taking the ratios we get the condition
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

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