$2\times$ distance to a line${} = {}$distance to a point, in geogebra

Question

While preparing for my math exams, I got this question: give the locus (if thats the right word) where $2$ times the distance to the line l $x=8$ equals $1$ time the distance to point F $2,0$. I was wondering how to get this shape visual in geogebra.

$l: x=8$

$F: (2,0)$

$P: d(P,F)=2*d(P,l)$, where d stands for distance

Answer 1

Shift down to $F=(0,0)$, the origin, and let $P= (x,y)$. To find $d(P,F)$, we simply have pythagorean theorem.

$$ d(P,F) = \sqrt{x^2 + y^2} $$

To find $d(P,l)$ where the line is $x=6$, we have that

$$ d(P,l) = |x-6| $$

If we require that $d(P,F) = 2 d (P,l)$ we see we just want points that satisfy

$$ x^2 +y^2 =4 |x-6|^2 $$

Thus we can simply graph $y^2 = 4|x-6|^2 -x^2$ and shift it over.

Answer 2

You wanted to visualize this with Geogebra. This is what I got provided I understand your problem correctly.

where A is the point (0,2) and the vertical line is x = 8.