One equation of a pair of dependent

(d) Condition for dependent linear equations

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}=\frac{1}{k}$ ...(i)

Given equation of line is, $-5 x+7 y-2=0$

Here, $a_{1}=-5, b_{1}=7, c_{1}=-2$

From Eq. (i), $-\frac{5}{a_{2}}=\frac{7}{b_{2}}=-\frac{2}{c_{2}}=\frac{1}{k}$ [say]

$\Rightarrow$ $a_{2}=-5 k, b_{2}=7 k, c_{2}=-2 k$

where, $k$ is any arbitrary constant.

Putting $k=2$, then $\quad a_{2}=-10, b_{2}=14$

and $c_{2}=-4$

$\therefore$ The required equation of line becomes

$a_{2} x+b_{2} y+c_{2}=0$

$\Rightarrow \quad-10 x+14 y-4=0$

$\Rightarrow \quad 10 x-14 y+4=0$