How do you find the value of a given the points (4,-1), (a,5) with a distance of 10?

Apr 11, 2018

color(blue)(a = 12, -4

Explanation:

Distance between two points is given by the formula :

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$\text{Given " (x_1, y_1) = (4,-1), (x_2,y_2) = (a,5), d = 10, " To find } a$

$10 = \sqrt{{\left(a - 4\right)}^{2} + {\left(5 + 1\right)}^{2}}$

${\left(a - 4\right)}^{2} + {6}^{2} = {10}^{2} , \text{ squaring both sides}$

${\left(a - 4\right)}^{2} = {10}^{2} - {6}^{2} = 64 = {8}^{2}$

$a - 4 = \pm 8 , \text{ taking root on both sides}$

color(brown)(a =) +-8 + 4 = color(brown)(12, -4

Apr 11, 2018

$a = 12 , a = - 4$

Explanation:

First, let's take a look at the distance formula where $d$ is the distance

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

Now you choose which point is point 2 (includes ${y}_{2}$ and ${x}_{2}$) and which point is point 1 (includes ${y}_{1}$ and ${x}_{1}$)

Point 2: $\left(a , 5\right) \to {y}_{2} = 5$ and ${x}_{2} = a$
Point 1: $\left(4 , - 1\right) \to {y}_{1} = - 1$ and ${x}_{1} = 4$

Now plug these values into the equation

$d = \sqrt{{\left(a - 4\right)}^{2} + {\left(5 - \left(- 1\right)\right)}^{2}}$

We also know that the distance $d$ is 10 so we can plug that in

$10 = \sqrt{{\left(a - 4\right)}^{2} + {\left(5 - \left(- 1\right)\right)}^{2}}$

Simplify

$10 = \sqrt{{\left(a - 4\right)}^{2} + {\left(5 - \left(- 1\right)\right)}^{2}}$

$10 = \sqrt{{\left(a - 4\right)}^{2} + {\left(5 + 1\right)}^{2}}$

$10 = \sqrt{{\left(a - 4\right)}^{2} + {\left(6\right)}^{2}}$

$10 = \sqrt{{\left(a - 4\right)}^{2} + 36}$

Now square both sides to get rid of the radical

${10}^{2} = {\left(\sqrt{{\left(a - 4\right)}^{2} + 36}\right)}^{2}$

$100 = {\left(a - 4\right)}^{2} + 36$

FOIL the expression ${\left(a - 4\right)}^{2}$

${\left(a - 4\right)}^{2}$

$\left(a - 4\right) \left(a - 4\right)$

${a}^{2} + a \left(- 4\right) + a \left(- 4\right) + \left(- 4\right) \left(- 4\right)$

${a}^{2} - 4 a - 4 a + 16$

${a}^{2} - 8 a + 16$

Now plug this expression back in for ${\left(a - 4\right)}^{2}$

$100 = {\left(a - 4\right)}^{2} + 36$

$100 = {a}^{2} - 8 a + 16 + 36$

$100 = {a}^{2} - 8 a + 52$

Subtract 100 from both sides

${a}^{2} - 8 a + 52 - 100 = 100 - 100$

${a}^{2} - 8 a - 48 = 0$

Now factor this expression to find the roots/zeros

${a}^{2} - 8 a - 48 = 0$

$\left(a - 12\right) \left(a + 4\right) = 0$

So the roots are

$a = 12 , a = - 4$