If the distance between the points $(2,\ -2)$ and $(-1,\ x)$ is $5$, find the values of $x$.

Given: Distance between the points $(2,\ -2)$ and $(-1,\ x)$ is $5$.

To do: To find the values of $x$.

Solution:

As given, $x_1=2,\ y_1=-2,\ x_2=-1,\ y_2=x$

Using the distance formula, Distance between the points$=\sqrt{( x_2-x_1)^2+( y_2-y_1)^2}$

$\Rightarrow 5=\sqrt{( -1-2)^2+( x-( -2))^2}$

$\Rightarrow 5=\sqrt{( -3)^2+( x+2)^2}$

$\Rightarrow 25=9+( x+2)^2$

$\Rightarrow ( x+2)^2=25-9$

$\Rightarrow ( x+2)^2=16$

$\Rightarrow ( x+2)=\pm\sqrt{16}$

$\Rightarrow ( x+2)=\pm4$

If $x+2=4\ \Rightarrow x=4-2=2$

If $x+2=-4\ \Rightarrow x=-4-2=-6$

Thus, $x=2,\ -6$.

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Updated on: 10-Oct-2022

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