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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.)

 Apr 9, 2021
 #1
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We  have  a circle  with a center  of  (-2, 7)   and a radius  of  13.......the  equation is

 

(x + 2)^2  + (y - 7)^2   =   13^2

 

Manipulating the equation of the line   we  have  that  x  = 2y + 10

Sub  this  into  the  circle equation 

 

(2y + 10 + 2)^2  +  ( y - 7)^2  =   13^2

 

(2y + 12)^2  +  ( y - 7)^2  =   169

 

4y^2  + 48y  + 144  + y^2  - 14y  +  49  = 169

 

5y^2 + 34y  + 24  =  0          factor

 

(5y  + 4) ( y + 6)   =  0

 

Set  each  factor  to  0  and  solve  for  y

 

5y + 4  =  0                                 y  +  6   =   0

 

5y  =  - 4                                    y  =  - 6

 

y  =  - 4/5

 

And

 

x = 2 (-4/5) + 10                             x  = 2(-6) + 10

 

x =  -8/5  + 50/5                              x  =   -2

 

x =  42/5

 

 

The  intersection points  are  (-2, - 6)   and  (42/5,  -4/5)

 

 

cool cool cool

 Apr 9, 2021
edited by CPhill  Apr 9, 2021

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