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 11th assignment.   

 

 

Problem 3

 

Investigate with different values of d

             

          

 

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

 

Now letís 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 ŰdŰ is the distance between the focus at the pole and its corresponding directrix.

 

Next letís begin our graphical reasoning.

 

 

 

What happens?

When we investigate with different positive values of d

 

 

Now letís use Graph Calculator to graph the polar equations.

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

 

Horizontal directrix below the pole: (see fig. 3)

 

Figure 3

 

 

Horizontal directrix above the pole: (see fig. 4)

 

 

 

Figure 4

 

 

Until now we can see how the shape of parabola changes as the value of d changes.

             

 

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

 

 

Horizontal directrix below the pole: (see fig. 7)

 

 

Figure 7

 

 

Horizontal directrix above the pole: (see fig. 8)

 

Figure 8

 

Until now we can see how the shape of ellipse changes as the value of d changes.

 

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

 

 

Horizontal directrix below the pole: (see fig. 11)

 

 

 

Figure 11

 

 

Horizontal directrix above the pole: (see fig. 12)

 

 

Figure 12

 

Until now we can see how the shape of hyperbola changes as the value of d changes.

 

 

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