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Coordinates

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142
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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.)

Jun 12, 2021

#1
+1

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)   Jun 12, 2021