SOLUTION: The point (a,b) is 5 units away from the point (6,3), and (a,b) lies on the line 5x - 4y = -14. What is the largest possible value of a?

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: The point (a,b) is 5 units away from the point (6,3), and (a,b) lies on the line 5x - 4y = -14. What is the largest possible value of a?      Log On


   

Question 1024962: The point (a,b) is 5 units away from the point (6,3), and (a,b) lies on the line 5x - 4y = -14. What is the largest possible value of a?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There must be a less painful way to do this.

The distance from (a,b) to (6,3) is
sqrt%28%28a-6%29%5E2%2B%28b-3%29%5E2%29=5 , so
%28a-6%29%5E2%2B%28b-3%29%5E2=25

5a-4b=14 because point (a,b) ies on the line 5x+-+4y+=+-14 .
5a-4b=14 --> b=%285%2F4%29a%2B7%2F2 --> b-3=%285%2F4%29a%2B1%2F2 --> %28b-3%29%5E2=%2825%2F16%29a%5E2%2B%285%2F4%29a%2B1%2F4

Substituting that expression in %28a-6%29%5E2%2B%28b-3%29%5E2=25 we get
%28a-6%29%5E2%2B%2825%2F16%29a%5E2%2B%285%2F4%29a%2B1%2F4=25
a%5E2-12a%2B36%2B%2825%2F16%29a%5E2%2B%285%2F4%29a%2B1%2F4=25
Multiplying both sides of the equal sign times 16 , we get
16a%5E2-192a%2B576%2B25a%5E2%2B20a%2B4=400-->41a%5E2-172a%2B180=0
Applying the quadratic formula, we get
a+=+%28-%28-172%29+%2B-+sqrt%28%28-172%29%5E2-4%2A41%2A180+%29%29%2F%282%2A41%29+
a+=+%28172+%2B-+sqrt%2829584-29520%29%29%2F82
a+=+%28172+%2B-+sqrt%2864%29%29%2F82
a+=+%28172+%2B-+8%29%2F82--->