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 information from the problem and solving for #a# gives:
#sqrt(50) = sqrt((color(red)(a) - color(blue)(-4))^2 + (color(red)(8) - color(blue)(1))^2)#
#sqrt(50) = sqrt((color(red)(a) + color(blue)(4))^2 + (color(red)(8) - color(blue)(1))^2)#
#sqrt(50) = sqrt((color(red)(a) + color(blue)(4))^2 + 7^2)#
#sqrt(50) = sqrt(a^2 + 8a + 16 + 49)#
#sqrt(50) = sqrt(a^2 + 8a + 65)#
#(sqrt(50))^2 = (sqrt(a^2 + 8a + 65))^2#
#50 = a^2 + 8a + 65#
#50 - color(red)(50) = a^2 + 8a + 65 - color(red)(50)#
#0 = a^2 + 8a + 15#
#0 = (a + 3)(a + 5)#
Solution 1:
#a + 3 = 0#
#a + 3 - color(red)(3) = 0 - color(red)(3)#
#a + 0 = -3#
#a = -3#
Solution 2:
#a + 5 = 0#
#a + 5 - color(red)(5) = 0 - color(red)(5)#
#a + 0 = -5#
#a = -5#
The Solution Is:
#color(red)(a)# can be either #-3# or #-5# for the two points to have a distance of #sqrt(50)#
