Answer
$k $ = $ \frac{1}{2}$ OR $0.5$
Work Step by Step
Distance 'd' between two points $(x_{1},y_{1}) $ and $(x_{2},y_{2})$ is given by-
d = $\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}} $
Now, Distance '$d_{1}$' between points $(1,k) $and $(0,0)$ will be -
$d_{1}$ = $\sqrt{(1 - 0)^{2} + (k - 0)^{2}} $
i.e. $d_{1}$ = $\sqrt{1 +k^{2} } $
Now, Distance '$d_{2}$' between points $(1,k) $and $(2,1)$ will be -
$d_{2}$ = $\sqrt{(1 - 2)^{2} + (k - 1)^{2}} $
i.e. $d_{2}$ = $\sqrt{(-1)^{2} + (k - 1)^{2}} $
i.e. $d_{2}$ = $\sqrt{1 + (k - 1)^{2}} $
i.e. $d_{2}$ = $\sqrt{1 + k^{2} +1 -2k} $
i.e. $d_{2}$ = $\sqrt{k^{2} -2k +2} $
Now According to problem-
$d_{1}$ = $d_{2}$
i.e. $\sqrt{1 +k^{2} } $ = $\sqrt{k^{2} -2k +2} $
Squaring on both sides-
$1 +k^{2} $ = $k^{2} -2k +2$
i.e. $1 $ = $ -2k +2$ (Subtracting $k^{2}$ from both sides)
i.e. $2k $ = $ 2-1 =1$
i.e. $k $ = $ \frac{1}{2}$ OR $0.5$
