Question

The combined equation of the straight lines of the form $y=k x+1$ (where $k$ is an integer) such that the point of intersection of each with the line $3 x+4 y=9$ has an integer as its $x$ - coordinate is

• A. $(y+x+1)(y+2 x-1)=0$
• B. $(y+x-1)(y+2 x+1)=0$
• C. $(y+x+1)(y+2 x+1)=0$
• D. $(y+x-1)(y+2 x-1)=0$

Answer 1

Answer: (D)

Given,
$y =k x+1\,\,\,...(i)$
and $ 3 x+4 y =9\,\,\,...(ii) $
Put the value of $y$ by Eq. (i) in Eq (ii), we get
$3 x+4(k x+1)=9$
$3 x+4 k x+4=9$
$\Rightarrow x(3+4 k)+4-9=0$
$\Rightarrow x(3+4 k)-5=0$
$\Rightarrow x(3+4 k)=5$
$\Rightarrow x=\frac{5}{3+4 k}$
$Sx=\pm 1, \pm 5 \,\,\,[\because x$ is an integer $]$
$ \therefore 3+4 k =\pm 1 $ or $ 3+4 k=\pm 5 $
$ k =\frac{-1}{2},-1,-2, \frac{1}{2} $
$\therefore $ we can choose value of $k$ only integer number.
$\therefore k=-1,-2$
Hence, combined equation is
$\{y=(-1) x+1\}\{y=(-2) x+1\}=0 $
$\Rightarrow (y=-x+1)(y=-2 x+1)=0 $
$\Rightarrow (x+y-1)(2 x+y-1)=0$