The nice story about Parabella, from Advanced Topics in Geometry, Conic4  PP1012, can continue, because she is more beautiful than Elli and Hyp think. It's not true that she has only one eye (focus). She too has two eyes as well, but her second eye is located at the infinity. Parabella is very farsighted. With her second eye she watches the stars as if she was on them. That's how she helps astronomers bringing space information to their telescopes, and sheds out the cars' headlights. If you move one focus of any ellipse or hyperbola infinitely far, they will turn into parabola automatically. Not just different parabolas. There is only ONE shape of parabola. All other parabolas are the same parabola just magnified or reduced in size differently. Like a circle: There is only one circle and all others are magnified or reduced versions of it. The same is with parabola. But it sounds strange. The reason is: they are chopped at different places. In fact parabola is infinite. We draw them chopped differently to fit on paper. If we chop the circle into different arcs they also will look like different shapes. All circles are similar to each other. You can chose the center of similarity at the center of the circle and enlarge it from that center radially. The same is true for parabola. Chose the center of similarity at the focus of the parabola and enlarge it radially from that focus. Or chose the vertex as center of similarity. We will get all other possible parabolas, which are all similar to each other. Whereas there are many different shapes for ellipses and hyperbolas, more elongated, or roundish... Because of this property of parabola her focus has a distinctive location. Draw from the focus the focal line parallel to the directrix. It intersects the parabola at two points. Chose one of them and drop from it a perpendicular to the directrix. You will get a SQUARE. This is true for any parabola. The square is also unique, like a circle, all possible squares are similar to each other they are enlarged or reduced versions. Because the square is unique the related parabola is unique too. One can use a similarity transformation to show that all parabolas are similar. The picture below compares the SIMILAR fragments of two different parabolas. This focal square is what describes the parabola in terms of her equation. The side of that square is 2p. The focal width of parabola is 4p. Now we see why x=2p gives y=p from the equation of parabola. Beautiful and unique focal square.


Exercise: Click here to play 'DRAW PARABOLA' game. A parabola is drawn on the paper with its axis of symmetry. No focus and no directrix are shown. How to find its focus? Answer: Draw a line from the vertex with the slope = 1/2. It will intersect parabola at the height of the focus. 