Question

If a line is equally inclined with the co-ordinate axes and also equally inclined with $2x-y+7=0$ and $ky-k^{2}x+y-2=0$, then the possible number of value(s) of $k$ is

Answer 1

The slope of the line which is equally inclined with co-ordinate axes is
$m=\pm 1$

Given lines are $2x-y+7=0$ and $ky-k^{2}x+y-2=0$
Here, the slope is
$M_1=2,M_2=\frac{k^2}{k+1}$

If two lines with slope $M_1$ and $M_2$ are equally inclined to a line with slope $m$, then
$\left(\frac{M_1-m}{1+M_{1}m}\right)=-\left(\frac{M_2-m}{1+M_{2}m}\right)$

When $m=1$, we get
$-\left(\frac{2-1}{1+2}\right)=\left(\frac{\frac{k^2}{k+1}-1}{1+\frac{k^2}{k+1}}\right)\Rightarrow -\frac{1}{3}=\frac{k^2-k-1}{k^2+k+1}\Rightarrow 2k^2-k-1=0\Rightarrow (2k+1)(k-1)=0\Rightarrow k=-\frac{1}{2},1$

When $m=-1$, we get
$-\left(\frac{2+1}{1-2}\right)=\left(\frac{\frac{k^2}{k+1}+1}{1-\frac{k^2}{k+1}}\right)\Rightarrow 3=\frac{k^2+k+1}{k+1-k^2}\Rightarrow 2k^2-k-1=0\Rightarrow k=-\frac{1}{2},1$

Hence, there exist $2$ values of $k$.