Why does this would eventually simplify into the original circle equation? I was trying to solve this problem:
The point $A$ has coordinates $(5, 16)$ and the point $B$ has coordinates $(-4,4)$. The variable $P$ has coordinate $(x,y)$ and moves on a path such that $ AP = 2BP$. Show that the Cartesian equation of the path of the $P$ is:
$$(x+7)^2 +y^2 = 100\tag{*}$$
So, what I did is to found a point on the circular path, and shows that the relationship holds. However, the actual answer is to let:
$$(x-5)^2 +(y-16)^2 = 4(x+4)^2 +4(y-4)^2$$
This also just means $AP=2BP$.
However, it will simplify to the original circle equation $(*)$. I literally don't know why. Even if $AP=2BP$ , why would this simplify to $(*)$. What's the mechanism behind this.  
Thank you very much for you guys reply. 

The locus of the points such that the ratio of their distance to two given points is constant is a circle:
$$\frac{\sqrt{(x-x_1)^2+(y-y_1)^2}}{\sqrt{(x-x_0)^2+(y-y_0)^2}}=\lambda.$$
This is because
$$(x-x_1)^2+(y-y_1)^2-\lambda^2((x-x_0)^2+(y-y_0)^2)=0$$ is the equation of a conic, such that 


*

*the coefficients of $x^2,y^2$ are equal, and

*with no cross term $xy$.
These are the conditions to have a circle.
When $\lambda=1$, the quadratic terms cancel each other, giving the equation of a line (the mediatrix).
