If the distance between the points $ \mathrm{A}(a, 2) $ and $ \mathrm{B}(3,-5) $ is $ \sqrt{53} $, find the possible values of $ a $.

Given:

The distance between the points \( \mathrm{A}(a, 2) \) and \( \mathrm{B}(3,-5) \) is \( \sqrt{53} \).

To do:

We have to find the value of $a$.

Solution:

We know that,

The distance between two points \( \mathrm{A}\left(x_{1}, y_{1}\right) \) and \( \mathrm{B}\left(x_{2}, y_{2}\right) \) is \( \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \).

Therefore,

 The distance between the points $(a, 2)$ and $(3, -5)$ is,

$\sqrt{53}=\sqrt{(3-a)^2+(-5-2)^2}$

Squaring on both sides, we get,

$(\sqrt{53})^2=(\sqrt{(3-a)^2+(-7)^2})^2$

$53=9+a^2-6a+49$

$a^2-6a+58-53=0$

$a^2-6a+5=0$

$a^2-5a-a+5=0$

$a(a-5)-1(a-5)=0$

$(a-5)(a-1)=0$

$a-5=0$ or $a-1=0$

$a=5$ or $a=1$

The value of $a$ is $1$ or $5$.