If the distance between the points (4, 𝑘) and (1, 0) is 5, then what can be the possible values of 𝑘?

We have been given two points (4, 𝑘) and (1, 0) which are separated by a distance of 5 units and we need to find the value(s) of 𝑝.

We know that, distance between two points 𝑃 and 𝑄 are calculated by

 $PQ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$, Where  (𝑥₁ , 𝑦₁ ) are coordinates of point P and (𝑥₂, 𝑦₂) are coordinates of point 𝑄.

In the given case, (𝑥 1 , 𝑦 1 ) = (4, 𝑘) 𝑎𝑛𝑑 (𝑥 2 , 𝑦 2 ) = (1, 0)

Putting the values of coordinates in the formula, we get

$\begin{array}{r}5=\sqrt{(1-4{)}^2+(0-k{)}^2}\\ \Rightarrow 5=\sqrt{(-3{)}^2+(-k{)}^2}\end{array}$

$\begin{array}{r}\Rightarrow 25=(-3{)}^2+(k{)}^2\\ \Rightarrow 25=9+k^2\\ \Rightarrow k^2=16\\ \Rightarrow k=\pm 4\end{array}$

Hence, the values of 𝑘 are ± 4.