I made these remarkable "discoveries" while in college.

If you have a table lamp with a cylinderical shade,

turn it on and look at the pattern on the nearby wall.

You will see a.hyperbola

I was in the cafeteria, staring into my coffee mug.

There was particularly bright source of light nearby.

I saw what looked like a cardioid on the surface.

My friend and I spent the next hour proving it.

(We missed the next class.)

After working with [tex]x^2 \,=\,4py[/tex] for weeks, it finally

occured to me that a parabola hasparameter.one

Other than orientation, location and scale,

there is exactlyparabola. .How can this be?one

Aren't these two different parabolas?

Answer: No.Code:Fig. 1 Fig. 2 | | | * | * | | * | * * | * * | * * | * * | * * | * - - - - - - - * - - - - - - - - - - - - - * - - - - - - | |

Fig. 1 is an enlargement (close-up view) of Fig. 2.

There is one conic curve.

Consider the distance [tex]d[/tex] between the two foci.

If [tex]d = 0[/tex], we have a.circle

Code:* * * * | * * | * * | * | * F1| * * - - - - o - - - - * * |F2 * | * | * * | * * | * * * *

If [tex]d[/tex] is finite and nonzero, we have an.ellipse

Code:| * * * * | * * | * * | * | * F1| F2 * * - - - o - - - o - - - * * | d * | * | * * | * * | * | * * *

If [tex]d = \infty[/tex], we have a.parabola

Code:| * | * * | * | * | F1| * - - - o - - - - - - F2 → → | * | * | * | | * | *

Brace yourself!

If [tex]\color{purple}{d > \infty}[/tex], we have a.hyperbola

Code:| * | * * * * * | * * | → → o - - * - - - * - - o F2 * * |F1 * * | * * * | * |

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