# How do you find a possible value for a if the points (a,-6), (-5,2) has a distance of d=10?

Feb 26, 2017

There are two possible values for $a$: $- 11$ or $1$

#### 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 in the problem gives:

$10 = \sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{2} - \textcolor{b l u e}{- 6}\right)}^{2}}$

$10 = \sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{2} + \textcolor{b l u e}{6}\right)}^{2}}$

${10}^{2} = {\left(\sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{2} + \textcolor{b l u e}{6}\right)}^{2}}\right)}^{2}$

$100 = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + {\left(\textcolor{red}{2} + \textcolor{b l u e}{6}\right)}^{2}$

$100 = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + {8}^{2}$

$100 = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + 64$

$100 - \textcolor{red}{64} = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + 64 - \textcolor{red}{64}$

$36 = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2} + 0$

$36 = {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2}$

$\sqrt{36} = \sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{a}\right)}^{2}}$

$6 = \pm \left(- 5 - a\right)$

Solution 1)

$6 = - \left(- 5 - a\right)$

$6 = 5 + a$

$- \textcolor{red}{5} + 6 = - \textcolor{red}{5} + 5 + a$

$1 = 0 + a$

$1 = a$

$a = 1$

Solution 2)

$6 = - 5 - a$

$\textcolor{red}{5} + 6 = \textcolor{red}{5} - 5 - a$

$11 = 0 - a$

$11 = - a$

$\textcolor{red}{- 1} \times 11 = \textcolor{red}{- 1} \times - a$

$- 11 = a$

$a = - 11$

The solutions are $a = - 11$ or $a = 1$