Question 655501
<pre>
To avoid doing people's homework, I'll work a problem exactly like yours,
but with different numbers.
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Find the coordinates of P if P is 17 units from L(-3,-6) and 10 units from M(-1,1)

It appears that there are two solutions.

{{{drawing(4200/11,700,-5,7,-9,13,

graph(4200/11,700,-5,7,-9,13), locate(-4,-6-.2,"L(-3,-6)"), circle(-3,-6,.1),

locate(-1-.9,1-.1,"M(-1,1)"), circle(-1,1,.1), 

red(locate(5,9+.5,"P(x,y)"), circle(5,9,.1)), 

green(line(5,9,-3,-6)), red(line(5,9,-1,1)) )}}}OR{{{drawing(4200/11,700,-5,7,-9,13, locate(-1-.9,1-.1,"M(-1,1)"),

graph(4200/11,700,-5,7,-9,13), locate(-4,-6-.2,"L(-3,-6)"), circle(-3,-6,.1),

 

red(locate(-99/53-.8,581/53+.5,"P(x,y)"), circle(-99/53,581/53,.1)), 

green(line(-99/53,581/53,-3,-6)), red(line(-99/53,581/53,-1,1)) )}}}

We use the distance formula, setting PL = 17 and PM = 10

d = {{{sqrt((x[2]-x[1])^2+(y[2]-y[1])^2)}}}

{{{sqrt(x+3)^2+(y+6)^2)}}} = 17
{{{sqrt(x+1)^2+(y-1)^2)}}} = 10

Squaring both sides of both equations:

(x+3)² + (y+6)² = 17²
(x+1)² + (y-1)² = 10²

x²+6x+9 + y²+12y+36 = 289
x²+2x+1 +   y²-2y+1 = 100

Simplifying

x²+y²+6x+12y = 244
x²+y²+2x- 2y =  98

Subtracting those two equations:

      4x+14y = 146

Divide that equation through by 2

      2x+ 7y = 73
          2x = 73-7y
           x = {{{(73-7y)/2}}}

Substitute in

x²+y²+2x- 2y =  98
{{{((73-7y)/2)^2}}} + y² + 2{{{((73-7y)/2)}}} - 2y = 98

{{{((5329-1022y+49y^2)/4)}}} + y² + 73 - 7y - 2y = 98

{{{((5329-1022y+49y^2)/4)}}} + y² - 9y  = 25

Clear the fraction by multiplying through by 4

5329 - 1022y + 49y² + 4y² - 36y = 100

            53y² - 1058y + 5229 = 0

That factors as

             (53y - 581)(y - 9) = 0

           53y - 581 = 0;          y - 9 = 0
                 53y = 581;            y = 9
                   y = {{{581/53}}}

Substitute each in 2x + 7y = 73

             2x + 7y = 73
2x + 7({{{581/53}}}) = 73
         106x + 4067 = 3869
                106x = -198
                   x = {{{-198/106}}}
                   x = {{{-99/53}}}

So one solution is the point P({{{-99/53}}}, {{{581/53}}})

[That's approximately the point P(-1.97,10.96)

The other solution is

             2x + 7y = 73
           2x + 7(9) = 73
             2x + 63 = 73
                  2x = 10
                   x = 5

So the other solution is P(5,9)

Edwin</pre>