Question

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

Answer 1

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

Explanation:

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 in the problem gives:

`10 = sqrt((color(red)(-5) - color(blue)(a))^2 + (color(red)(2) - color(blue)(-6))^2)`

`10 = sqrt((color(red)(-5) - color(blue)(a))^2 + (color(red)(2) + color(blue)(6))^2)`

`10^2 = (sqrt((color(red)(-5) - color(blue)(a))^2 + (color(red)(2) + color(blue)(6))^2))^2`

`100 = (color(red)(-5) - color(blue)(a))^2 + (color(red)(2) + color(blue)(6))^2`

`100 = (color(red)(-5) - color(blue)(a))^2 + 8^2`

`100 = (color(red)(-5) - color(blue)(a))^2 + 64`

`100 - color(red)(64) = (color(red)(-5) - color(blue)(a))^2 + 64 - color(red)(64)`

`36 = (color(red)(-5) - color(blue)(a))^2 + 0`

`36 = (color(red)(-5) - color(blue)(a))^2`

`sqrt(36) = sqrt((color(red)(-5) - color(blue)(a))^2)`

`6 = +-(-5 - a)`

Solution 1)

`6 = -(-5 - a)`

`6 = 5 + a`

`-color(red)(5) + 6 = -color(red)(5) + 5 + a`

`1 = 0 + a`

`1 = a`

`a = 1`

Solution 2)

`6 = -5 - a`

`color(red)(5) + 6 = color(red)(5) - 5 - a`

`11 = 0 - a`

`11 = -a`

`color(red)(-1) xx 11 = color(red)(-1) xx -a`

`-11 = a`

`a = -11`

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