Hello, sux@math!

A dog walking through a yard. Tree $\displaystyle A(-2,3)$, Tree $\displaystyle B (4, -6)$

Dog is at all times, twice as far from $\displaystyle A$ as it is from $\displaystyle B. $

Draw a graph showing the trees, path of dog and relationship defining the locus.

Write geometric description of the path of the dog, relative to the two trees. Code:

A | D
* - + - - - - - - * (x,y)
(-2,3) | /
--------+-----------/--------
| /
| /
| /
| * (4,-6)
| B

The trees are at $\displaystyle A\text{ and }B$; the dog is at $\displaystyle D(x,y)$

Since $\displaystyle DA \:=\:2\!\cdot\!DB$, we have: .$\displaystyle \sqrt{(x+2)^2+(y-3)^2} \;=\;2\cdot\sqrt{(x-4)^2+(y+6)^2} $

Square both sides: .$\displaystyle x^2 + 4x + 4 + y^2 - 6y + 9 \;\;=\;\;4(x^2-8x+16 + y^2 + 12y + 36)$

. . which simplifies to: .$\displaystyle x^2 - 12x + y^2 + 18y \;=\;65$

Complete the square: .$\displaystyle (x^2-12x \;{\color{blue}+\: 36}) + (y^2 + 18y\;{\color{red}+\: 81}) \;=\;65 \;{\color{blue}+\: 36}\:{\color{red} +\: 81}$

. . and we have: .$\displaystyle (x-6)^2 + (y+9)^2 \;=\;182$

The dog's path is a circle with center $\displaystyle (6,\:\text{-}9)$ and radius $\displaystyle \sqrt{182}$