Definition of the Ellipse: If $M$ is a fixed point (the focus) and $T$ is a fixed straight line (the directrix), and $P$ is a moving point which moves in a certain way to maintain the ratio $\frac {PM}{PN}= e$ (where $0<e<1$, and $PN$ is the perpendicular from $P$ to the directrix), then the locus of the $P$ will be called an ellipse.
If we know only this definition of the ellipse, then how can we perceive the diagram of the ellipse? How can we say that the locus of the ellipse will be a closed curve?
Can anyone please help me?
We can check, for instance, if a ray issued from focus $M$ intersects the curve at a certain point $P$. Let $r$ be the distance $PM$, $a$ the distance from $M$ to the directrix $T$, $\theta$ the angle formed by ray $MP$ with the ray from $M$ perpendicular to $T$ but not intersecting $T$. The distance from $P$ to $T$ is then $r\cos\theta+a$and we have: $$ {r\over r\cos\theta+a} = e, \quad\text{that is:}\quad r={ae\over1-e\cos\theta}. $$ This gives a positive value of $r$ for any value of $\theta$, that is: every ray from the focus intersects the ellipse at a single point, distance $r$ is limited (${ae\over1+e}\le r\le{ae\over1+e}$) and $r(\theta)$ is a continuous function. It follows that the curve is closed.