One equation of a pair of dependent

Question:

One equation of a pair of dependent linear equations is – 5x+ 7y – 2 = 0. The second equation can be

(a) 10x + 14y + 4=0                                   

(b)-10x-14y + 4 =0

(c) -10x + 14y + 4 = 0                                

(d) 10x-14y + 4=0

Solution:

(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

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

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

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

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