A Calculus book that I'm self-studying is asking me to prove the following theorem about conic sections:
A conic section is symmetrical with respect to its principal axis.
Here is my attempt at a solution:
I will use the following definition of a conic section:
A conic section is the set of all points $P$ such that
$|\overline{FP}|=e|\overline{RP}|$,
where $P$ is a point in a plane, $e$ is the eccentricity, $F$ is the focus of the conic section and $R$ is the point in the directrix such that the line $\overline{RP}$ is perpendicular to the directrix. In other words, $|\overline{RP}|$ is the distance from $P$ to the directrix.
I will also use the following definitions:
The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}.$
The distance between a point $P(x,y)$ and the line $r: Ax+By+C=0$ is $|Ax+By+C| / \sqrt{A^2+B^2}$.
(Edit) The principal axis of a conic section is the line that goes through the focus $F$ and is perpendicular to the directrix.
For this solution, let's consider a conic section, and let $e$ be the eccentricity and $F$ be a focus of this conic section.
Let $d$ be the distance between the focus and the directrix. Let's place this conic section in a Cartesian coordinate system in such a way that:
1) the focus $F$ is in the origin, that is, $F$ is the point $(0,0)$;
2) the directrix is the line $x=-d$, or $x+d=0$.
In this particular configuration, the principal axis of the conic section is along the x-axis.
The value of $|\overline{FP}|$ is $\sqrt{(x-0)^2+(y-0)^2}=\sqrt{x^2+y^2}$. The value of $|\overline{RP}|$ is the value of the distance between point $P$ and the line $x+d=0$. Using the formula $|Ax+By+C| / \sqrt{A^2+B^2}$ with $A=1$, $B=0$ and $C=d$, we get $|\overline{RP}|=|x+d|$. Substituting these values in the definition of a conic section:
$|\overline{FP}|=e|\overline{RP}|$
$\sqrt{x^2+y^2}=e|x+d|$
Squaring both sides:
$x^2+y^2=e^2(x+d)^2$
$x^2+y^2=e^2 (x^2+2xd+d^2)$
Or:
$y=\pm \sqrt{e^2 (x^2+2xd+d^2)-x^2}$
The above result means that the graph of $y$ (that is, the conic section) is symmetrical with respect to the x-axis, and, therefore, the conic section is symmetrical with respect to the principal its axis. Is this correct or am I missing something?