Conic Sections

Caution: TEMATH's tools will write the values of computed results at the bottom of the Report window. This will cause you to constantly scroll between the computed results and the instructions for this activity. You may want to print a copy of the contents of the Report window before you begin this activity.

Exploring the Definition of the Ellipse

Let's use the ellipse

x^2/5^2 + y^2/3^2 = 1

for our exploration.

Select New Conic - Ellipse from the Work menu.

The ellipse template "e1: ellipse(x0,a,y0,b)" is written into the top cell of the Work window and the algebraic form of the ellipse "(x-x0)^2/a^2 + (y-y0)^2/b^2 = 1" is written at the bottom of this Report window.
Since the center of the ellipse (x0, y0) is at the origin (0, 0), a = 5 and b = 3, fill-in the template as "e1: ellipse(0,5,0,3)" and press the Enter key.
Select Plot from the Graph menu.

Note that solid circles are drawn at the locations of the foci and a circle is drawn at the center of the ellipse. Additionally, the coordinates of the foci and the eccentricity of the ellipse are written into this Report window. To control what conic annotations will be displayed with the plot, select Conic Annotations... from the Options menu.
Click the Line Segment tool on the tool palette.
Draw a line from the left focus point F1 to any point P on the ellipse. Draw a second line from the point P to the right focus point F2. The lengths of the two line segments, |F1P| and |F2P|, are written into TEMATH's Report window.

What is the sum of the lengths of the two line segments?

Conic Sections - Ellipse

Repeat this line drawing process for five other points on the graph of the ellipse and record the sums below:

Can you make any conclusion (conjecture) about the sums? (Take into account the fact that there is probably some measurement inaccuracy in drawing a line from a focus point to a point on the ellipse.)

Can you state your conclusion in terms of the parameter "a" from the standard ellipse equation?

Test your conclusion on other ellipses, for example, (x^2)/(4^2) + (y^2)/(2^2) = 1 and (x^2)/(3^2) + (y^2)/(4^2) = 1 . Is your conclusion valid for these ellipses?

How would you geometrically define an ellipse?

Exploring the Definition of the Hyperbola

Let's use the hyperbola

x^2/3^2 - y^2/5^2 = 1

for our exploration.

Select New Conic - Horizontal Hyperbola from the Work menu.
The hyperbola template "e#: hhyperb(x0,a,y0,b)" is written into the top cell of the Work window and the algebraic form of the hyperbola "(x-x0)^2/a^2 - (y-y0)^2/b^2 = 1" is written at the bottom of this Report window.
Since the center of the hyperbola (x0, y0) is at the origin (0, 0), a = 3 and b = 5, fill-in the template as "e#: hhyperb(0,3,0,5)" and press the Enter key.
Click in the Domain & Range window and change the plotting domain to –10 x 10.
Select Plot from the Graph menu.

Note that solid circles are drawn at the locations of the foci, a circle is drawn at the center of the hyperbola, and slant asymptotes are drawn. Additionally, the coordinates of the foci are written into this Report window.
If the Line Segment toolon the tool palette is not selected, click the Line Segment tool to select it.
Draw a line from the left focus point F1 to any point P on the hyperbola. Draw a second line from the point P to the right focus point F2. The lengths of the two line segments, |F1P| and |F2P|, are written into TEMATH's Report window.

What is the difference of the largest length minus the smallest length of these two line segments?

Plot of the hyperbola x^2/9 - y^2/5^2 = 1

Repeat this line drawing process for five other points on the graph of the hyperbola and record the differences below:

Can you make any conclusion (conjecture) about the differences? (Take into account the fact that there is probably some measurement inaccuracy in drawing a line from a focus point to a point on the hyperbola.)

Can you state your conclusion in terms of the parameter "a" from the standard hyperbola equation?

Test your conclusion on other hyperbolas, for example, (x^2)/(4^2) – (y^2)/(3^2) = 1 and (x^2)/(6^2) – (y^2)/(7^2) = 1. Is your conclusion valid for these hyperbolas?

How would you geometrically define a hyperbola?

Does your definition hold true for vertical hyperbolas? Try (y^2)/(3^2) – (x^2)/(2^2) = 1 and (y^2)/(5^2) – (x^2)/(7^2) = 1.

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