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 problem for the points and the distance and then solving for #a# gives:

#15 = sqrt((color(red)(a) - color(blue)(2))^2 + (color(red)(7) - color(blue)(-5))^2)#

#15 = sqrt((color(red)(a) - color(blue)(2))^2 + (color(red)(7) + color(blue)(5))^2)#

#15 = sqrt((color(red)(a) - color(blue)(2))^2 + 12^2)#

#15 = sqrt((color(red)(a) - color(blue)(2))^2 + 144)#

#15^2 = (sqrt((color(red)(a) - color(blue)(2))^2 + 144))^2#

#225 = (color(red)(a) - color(blue)(2))^2 + 144#

#225 = a^2 - 4a + 4 + 144#

#225 = a^2 - 4a + 148#

#225 - color(red)(225) = a^2 - 4a + 148 - color(red)(225)#

#0 = a^2 - 4a - 77#

#0 = (a - 11)(a + 7)#

Solution 1)

#a - 11 = 0#

#a - 11 + color(11) = 0 + color(11)#

#a - 0 = 11#

#a = 11#

Solution 1)

#a + 7 = 0#

#a + 7 - color(7) = 0 - color(7)#

#a + 0 = -7#

#a = -7#

#a# can be either #-7# or #11#