If the distance between the points $(4,\ p)$ and $(1,\ 0)$ is 5, then find the value of $p$.

Given: Distance between the points $(4,\ p)$ and $(1,\ 0)$ is $5$.

To do: To find the value of $p$.

Solution:

As given, $x_1=4,\ y_1=p,\ x_2=1,\ y_2=0$

Using the distance formula,

$5=\sqrt{( x_2-x_1)^2+( y_2-y_1)^2}$

$\Rightarrow 5=\sqrt{ (1-4)^2+( 0-p)^2}$

$\Rightarrow 5^2=( -3)^2+( -p)^2$

$\Rightarrow 25=9+p^2$

$\Rightarrow p^2=25-16$

$\Rightarrow p^2=9$

$\Rightarrow p=\sqrt{9}$

$\Rightarrow p=\pm3$

Therefore, the value of $p$ is $\pm3$.