Explore
Conics Family

Najia
Bao

When technological tools are available, students can
focus on decision making, reflection, reasoning, and problem solving (NCTM
2000).

Now we use Graph Calculator to
explore problem 3 in the 11^{th} assignment.

**Problem
3**

for e > 1, e = 1, e < 1.

Now let’s recall the concepts and theorems related to
polar equations of conics.

1.
The locus of a point in
the plane whose distance from a fixed point (focus) has a constant ratio to its
distance form a fixed line (directrix) is a conic. The constant ratio e is the
eccentricity of the conic.

2.
A polar equation of the
form or represents a conic
section with eccentricity e. The conic is an ellipse if e < 1, a parabola
if e =1, or a hyperbola if e > 1. And ôdô
is the distance between the focus at the pole and its corresponding directrix.

Next let’s begin our graphical reasoning.

**What happens?**

**When we investigate with different positive values
of d**

*For e = 1*, when d = 1, 2, 3, and 4 respectively, the graphs of
four types of equations shown in Figure 1, 2, 3, and 4 can be classified as
follows:

Vertical
directrix left the pole: (see fig. 1).

Figure 1

Vertical directrix right the pole: (see fig. 2)

Figure 2

Figure 3

Figure 4

*For e **<** 1*,
suppose e = 0.7, and when d = 1, 2, 3, and 4 respectively, the graphs of four
types of equations shown in Figure 1, 2, 3, and 4 can be classified as follows:

Vertical
directrix left the pole: (see fig. 5).

Figure 5

Vertical directrix right the pole: (see fig. 6)

Figure 6

Figure 7

Figure 8

*For e **>** **1*,
suppose e = 1.5, and when d = 1, 2, 3, and 4 respectively, the graphs of four
types of equations shown in Figure 1, 2, 3, and 4 can be classified as follows:

Vertical
directrix left the pole: (see fig. 9).

Figure 9

Vertical directrix right the pole: (see fig. 10)

Figure 10

Figure 11

Figure 12

**One Interesting and amazing phenomenon:**

**Two groups of curves are coincident**

** **

** **When we go on to investigate with different negative
values of d, we find an interesting and amazing phenomenon that the group of
the curves and the group of the curves are coincident (see fig. 13 & fig. 14). That is because
any point (-r, q) on the curve represents the
same point as any point (r, q + p) on the curve .

Figure 13

Figure 14

**Another Interesting and amazing phenomenon:**

**Two groups of curves are orthogonal**

When
we continue our exploration, we find another interesting and amazing
phenomenon, that is, two groups of curves and are
orthogonal (see fig. 15). As we
know, an orthogonal trajectory of a group of curves is a curve that intersects
each curve of the group orthogonally, that is, at right angles. We say that the
two groups are orthogonal trajectories of each other. This is really a strange
phenomenon. But it can be explained by differential equations. The first step
of the proof is to write a differential equation that is satisfied by all
members of the group. This means to find the slope of the tangent line at any
point on one of the group of curves. The second step is to solve the
differential equation. Due to the time limitation, the detail of proof we will
discuss next class. So much for today.

Figure 15