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

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Answer 1 · 2017-07-05T15:08:23

See a solution process below:

The formula for calculating the distance between two points is:

$d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)$

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

$5 = sqrt((color(red)(5) - color(blue)(a))^2 + (color(red)(-1) - color(blue)(3))^2)$

We can now solve for $a$ :

Squaring both sides of the equation gives:

$5^2 = (sqrt((color(red)(5) - color(blue)(a))^2 + (color(red)(-1) - color(blue)(3))^2))^2$

$25 = (color(red)(5) - color(blue)(a))^2 + (color(red)(-1) - color(blue)(3))^2$

$25 = (color(red)(5) - color(blue)(a))^2 + (-4)^2$

$25 = (color(red)(5) - color(blue)(a))^2 + 16$

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

$-color(red)(25) + 25 = -color(red)(25) + 25 - 10a + a^2 + 16$

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

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

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

$0 = (a - 8)(a - 2)$

$(a - 8)(a - 2) = 0$

Now, solve each term for $0$ :

Solution 1)

$a - 8 = 0$

$a - 8 + color(red)(8) = 0 + color(red)(8)$

$a - 0 = 8$

$a = 8$

Solution 2)

$a - 2 = 0$

$a - 2 + color(red)(2) = 0 + color(red)(2)$

$a - 0 = 2$

$a = 2$

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