circlesgeometry
I'm having some trouble with the following exercise:
Let $l$ be a line and let $\gamma$ be a circumference. If $\gamma \cap l = \emptyset$, what is the set of all points that are the same distance from $\gamma$ and $l$?
I draw a sketch and I think that it's a Parabola with $l$ being the directrix, but I can't find what the focus would be. Is this correct? If so, how Can I Prove it?
Suppose you have the focus at point $(a, b)$ and a directrix with equation $cx+dy=e$. Then a point $(x,y)$ is on the parabola if it has the same distance from focus and directrix, which means
$$(a-x)^2+(b-y)^2=\frac{(cx+dy-e)^2}{c^2+d^2}$$
You can rewrite that as
$$ (x,y,1) \begin{pmatrix} d^2 & -cd & ce-a(c^2+d^2) \\ -cd & c^2 & de-b(c^2+d^2) \\ ce-a(c^2+d^2) & de-b(c^2+d^2) & (a^2+b^2)(c^2+d^2) - e^2 \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} = 0 $$
If you choose the Hesse normal form for the line, with unit length normal vector, you have $c^2+d^2=1$ which will make the matrix easier to read. Note that the entries of this matrix correspond to the coefficients Awesome assumed in his comment.
Do this for both your parabolas, and you get two matrices which you can then feed into the generic method for intersecting conics.
PERHAPS THIS HELPS
I've been thinking about your problem for a while. I wouldn't say that I could solve it. However, I have an idea that I would like to share with you.
You certainly remember the Dandelin spheres which may have something to do with your conjecture. The following figure depicts two cones (a black one and a grey one). These cones share the vertical axis and a Dandelin sphere. Tangent to this sphere there is a blue plane whose edge can be seen in the figure:
The blue plane and the black cone intersect in a parabola. Also, the blue plane intersects with the other cone in a hyperbola. The parabola's focus and one of the foci of the hyperbola are common.
My conjecture is that this is how to imagine your couple of parabola and hyperbola with a common focus.
Then, I don't know how to proceed.
Best Answer
It is a parabola, with its focus at the center of circle $\gamma$, and its directrix parallel to $l$, at a distance from it equal to the radius of $\gamma$.