SOLUTION: P(2,0)and Q(0,-2)are two fixed points, find the equation of the locus of the moving point R such that PR:QR=1:2

Algebra ->  Length-and-distance -> SOLUTION: P(2,0)and Q(0,-2)are two fixed points, find the equation of the locus of the moving point R such that PR:QR=1:2      Log On


   

Question 1036348: P(2,0)and Q(0,-2)are two fixed points, find the equation of the locus of the moving point R such that PR:QR=1:2
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
This might be a slow, inefficient way to the solution,
but I cannot think of a better one right now.
There must be a physics (electricity and magnetism) simulation program that graphs this locus nicely, but I do not see how it would help.

Since P and Q are different points, R%28x%2Cy%29 is a third different point.
PR%2FQR=1%2F2 and we know that PR%3C%3E0 , so
PR%2FQR=1%2F2-->PR%5E2%2FQR%5E1=1%2F4<-->4PR%5E2=QR%5E2
Now we get to use x and y to find the equation representing the locus of point R%28x%2Cy%29 .
PR%5E2=%28x-2%29%5E2%2By%5E2
QR%5E2=x%5E2%2B%28y%2B2%29%5E2
4%28%28x-2%29%5E2%2By%5E2%29=x%5E2%2B%28y%2B2%29%5E2
4%28x%5E2-4x%2B4%2By%5E2%29=x%5E2%2By%5E2%2B4y%2B4
4x%5E2-16x%2B16%2B4y%5E2=x%5E2%2By%5E2%2B4y%2B4
3x%5E2-16x%2B3y%5E2-4y=4-16
3x%5E2-16x%2B3y%5E2-4y=-12
x%5E2-%2816%2F3%29x%2By%5E2-%284%2F3%29y=-4

%28x-8%2F3%29%5E2%2B%28y-2%2F3%29%5E2=-36%2F9%2B64%2F9%2B4%2F9
%28x-8%2F3%29%5E2%2B%28y-2%2F3%29%5E2=32%2F9
That is the equation of a circle,
centered at C%288%2F3%2C2%2F3%29 ,
with radius sqrt%2832%2F9%29=4sqrt%282%29%2F3 .