If the distance between the points (5, - 2) and (1, a) is 5 units then find the values of a.

We know, the distance between two points is  $\sqrt{{\left({\mathrm{x}}_1-{\mathrm{x}}_2\right)}^2+{\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)}^2}$

Where $(x_1, y_1) and (x_2,y_2)$coordinates of two points.

Here the distance = 5 units , x₁ = 5, x₂ = 1, y₁ = - 2 and y₂ = a

Therefore, on substitution:

 $\Rightarrow 5= \sqrt{{\left(5-1\right)}^2+{\left(-2-\mathrm{a}\right)}^2}$

 $\Rightarrow 25=16+{\left(2+\mathrm{a}\right)}^2$

 $\Rightarrow {\left(2+\mathrm{a}\right)}^2=25-16$

 $\Rightarrow {\left(2+\mathrm{a}\right)}^2=9$

Taking square root, 

$\Rightarrow 2+\mathrm{a}=\pm 3$

$\Rightarrow \mathrm{a}=-2\pm 3$

$\therefore a = 1, -5$