# How do you find the value of a given the points (2,-5), (a,7) with a distance of 15?

Jun 11, 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 problem for the points and the distance and then solving for $a$ gives:

$15 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{7} - \textcolor{b l u e}{- 5}\right)}^{2}}$

$15 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{7} + \textcolor{b l u e}{5}\right)}^{2}}$

$15 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + {12}^{2}}$

$15 = \sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + 144}$

${15}^{2} = {\left(\sqrt{{\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + 144}\right)}^{2}$

$225 = {\left(\textcolor{red}{a} - \textcolor{b l u e}{2}\right)}^{2} + 144$

$225 = {a}^{2} - 4 a + 4 + 144$

$225 = {a}^{2} - 4 a + 148$

$225 - \textcolor{red}{225} = {a}^{2} - 4 a + 148 - \textcolor{red}{225}$

$0 = {a}^{2} - 4 a - 77$

$0 = \left(a - 11\right) \left(a + 7\right)$

Solution 1)

$a - 11 = 0$

a - 11 + color(11) = 0 + color(11)

$a - 0 = 11$

$a = 11$

Solution 1)

$a + 7 = 0$

a + 7 - color(7) = 0 - color(7)

$a + 0 = - 7$

$a = - 7$

$a$ can be either $- 7$ or $11$