Find all points that are 13 units away from the point (-2,7) and that lie on the line x - 2y = 10. Give your answer as a list of points separated by semicolons, with the points ordered such that their x-coordinates are in increasing order. (So "(1,-3); (2,3); (5,-7)" - without the quotes - is a valid answer format.)
+129847 We have two equations
x - 2y = 10 ⇒ x = 2y + 10 (1)
(x +2)^2 + ( y -7)^2 = 169 (2)
Put (1) into (2)
(2y + 10 + 2)^2 + ( y - 7)^2 = 169
(2y + 12)^2 + (y - 7)^2 = 169
4y^2 + 48y + 144 + y^2 - 14y + 49 = 169
5y^2 + 34y + 193 = 169
5y^2 + 34y + 24 = 0 factor as
(5y + 4)(y + 6) = 0
Set each factor to 0 and solve for y and we have
y = -4/5 and y = -6
x = 2(-4/5)+ 10 x = 2(-6) + 10
x = 42/5 x = -2
The points are (-2, -6) and (42/5 -4/5)
