# How do you find the value of a given the points (a,3), (5,-1) with a distance of 5?

Jul 5, 2017

See a solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2}}$

Substituting the values from the points and for the distance in the problem gives:

$5 = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{3}\right)}^{2}}$

We can now solve for $a$:

Squaring both sides of the equation gives:

${5}^{2} = {\left(\sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{3}\right)}^{2}}\right)}^{2}$

$25 = {\left(\textcolor{red}{5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{3}\right)}^{2}$

$25 = {\left(\textcolor{red}{5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(- 4\right)}^{2}$

$25 = {\left(\textcolor{red}{5} - \textcolor{b l u e}{a}\right)}^{2} + 16$

$25 = 25 - 10 a + {a}^{2} + 16$

$- \textcolor{red}{25} + 25 = - \textcolor{red}{25} + 25 - 10 a + {a}^{2} + 16$

$0 = 0 - 10 a + {a}^{2} + 16$

$0 = - 10 a + {a}^{2} + 16$

$0 = {a}^{2} - 10 a + 16$

$0 = \left(a - 8\right) \left(a - 2\right)$

$\left(a - 8\right) \left(a - 2\right) = 0$

Now, solve each term for $0$:

Solution 1)

$a - 8 = 0$

$a - 8 + \textcolor{red}{8} = 0 + \textcolor{red}{8}$

$a - 0 = 8$

$a = 8$

Solution 2)

$a - 2 = 0$

$a - 2 + \textcolor{red}{2} = 0 + \textcolor{red}{2}$

$a - 0 = 2$

$a = 2$

The solution is: $a$ can be either $8$ or $2$