How do you find the value of a given the points (8,-5), (a,4) with a distance of #sqrt85#?

1 Answer
Nov 14, 2017

Answer:

See a solution process below:

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 and solveing for #a# gives:

#sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + (color(red)(4) - color(blue)(-5))^2)#

#sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + (color(red)(4) + color(blue)(5))^2)#

#sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + 9^2)#

#sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + 81)#

#(sqrt(85))^2 = (sqrt((color(red)(a) - color(blue)(8))^2 + 81))^2#

#85 = (color(red)(a) - color(blue)(8))^2 + 81#

#85 = a^2 - 16a + 64 + 81#

#85 - color(red)(85) = a^2 - 16a + 64 + 81 - color(red)(85)#

#0 = a^2 - 16a + 60#

#a^2 - 16a + 60 = 0#

#(a - 10)(a - 6) = 0#

Solution 1:

#a - 10 = 0#

#a - 10 + color(red)(10) = 0 + color(red)(10)#

#a - 0 = 10#

#a = 10#

Solution 2:

#a - 6 = 0#

#a - 6 + color(red)(6) = 0 + color(red)(6)#

#a - 0 = 6#

#a = 6#

#a# can be either #6# or #10#