Assignment Due 2/13

1. Answer Questions A on Lab Sheet 5.

2. Equations: Answer Question C3 on Lab Sheet 5 to give the equation for general e.

3. Parabolas

This problem refers to the parabola construction in Lab Sheet 5 and also in Investigation 1, Exploration 6.4 of GTC, pp. 89-90. We use the same notation as GTC. We are given a point A and a line BC. The locus is the locus of all the centers P of all the circles through A which are tangent to line BC.

1. Explain why the set of centers P is the same as the set of points that are equidistant from A and from line BC.
2. If the circle is tangent to line BC at point T, prove that the perpendicular bisector m of AT is the tangent to the parabola as outlined in the lab. (Note correction in the online version of the lab; corrected text is in red.)
3. The parabola has one line of symmetry. Tell what it is and why it is a line of symmetry. This line of symmetry is called the axis of the parabola. Explain why the axis is the only line (besides tangents) that intersects the parabola in exactly one point.
4. Use the distance formulas in the plane to find the equation for the set of all points P which are equidistant from the y-axis and point (p,0). Show how this equation relates to the equation in Problem 2, when e = 1.
5. Using the tangent line from (b) above, show that if a light ray parallel to the axis strikes the parabola at P, it is reflected to a ray through the focus F. Also, note that the reflection of F in any tangent line if a point F' on the directrix.

4A. Constructing Parabolas from two tangent points

Suppose that you know a point A and the tangent line a at A, and also a point B and the tangent line b at B. Let C be the intersection of a and b and let D the point so that ACBD is a parallelogram.

A conjecture (which will be supported in Monday's class) is that line CD is parallel to the axis of the parabola. Assuming this, use reflection property 3e above to construct the focus and directrix of the parabola.

Now with Sketchpad, draw two curves starting with A, C, B. Let a = line AC and let b = line CB. Then see whether they appear to be the same.

• Use the focus and directrix and "locus of circle centers" construction to construct the parabola with this focus and directrix.
• Carry out the Bezier construction from Lab 2 of Math 444, with A1 = A, A2 = C, A3 = B. (Note: To get the whole parabola, you can let B1 be a point on LINE A1A2 instead of segment A1A2)

4B. Ellipses from Circles

This problem refers to the ellipse/hyperbola construction from part E of Lab 5, also in Investigation 2, Exploration 6.4 of GTC, pp. 91-92. We use the notation of GTC.

• In the construction, if A is inside the circle with center B through C, then the locus of P is an ellipse. Prove this by showing that a sum of distances is constant
• Find the lines of symmetry of the ellipse and the points where the ellipse intersects a line of symmetry.
  

5. Hyperbolas from Circles

This problem refers to the ellipse/hyperbola construction from part E of Lab 5, also in Investigation 2, Exploration 6.4 of GTC, pp. 91-92. We use the notation of GTC.

• In the construction, if A is outside the circle with center B through C, then the locus of P is a hyperbola. Prove this by showing that a difference of distances has constant absolute value.
• Find the lines of symmetry of the hyperbola and the points where the hyperbola intersects a line of symmetry.
• For the hyperbola case, there are two positions of T for which P does not exist ("P goes to infinity"). What are these positions of T precisely and why does P not exist in these cases? (These are the asymptotes.)