Question

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

Answer 1

Slope of the line equally inclined with co-ordinate axes is $m=\pm 1$
Here $M_1=2,M_2=\frac{k^2}{k+1}$
If two lines with thw slope $M_1$ and $M_2$ are equally inclined to a line with slope $m$, then
$\left(\frac{M_1-m}{1+M_{1}m}\right)=-\left(\frac{M_2-m}{1+M_{2}m}\right)$
Case:1 (If $m=1$)
$-\left(\frac{2-1}{1+2}\right)=\left(\frac{\frac{k^2}{k+1}-1}{1+\frac{k^2}{k+1}}\right)$
$\Rightarrow -\frac{1}{3}=\frac{k^2-k-1}{k^2+k+1}$
$\Rightarrow 2k^2-k-1=0$
Two values of $k$ exist.

Case:2(If $m=-1$)
$-\left(\frac{2+1}{1-2}\right)=\left(\frac{\frac{k^2}{k+1}+1}{1-\frac{k^2}{k+1}}\right)$
$\Rightarrow 3=\frac{k^2+k+1}{-(k^2-k-1)}$
$\Rightarrow 2k^2-k-1=0$
$△>0$
Two values of $k$ exist.
$\therefore$only $2$ value's of $k$ exists as both equations represent same