Math 445 Assignments for Week 1

Reading Assignment with Problems, Due Friday, 1/5 (10 points -- see note)

Read BEG 1.1.1 and 1.1.2 and do the problems (Problems 1 through 6) for inspection at the beginning of class.

Assignment 1, Due Monday 1/8 (45 points)

Read: BEG 1.1.3 and 1.1.5 (discussed in class Friday, 1/5). Read Ogilvy, pp. 36-39 and pp. 73-86 (for discussion Monday, 1/8). Read Exploration 6.4 of GTC.

Reading Question: In the BEG text, the writers use the British usage of the word "gradient". In American English we use a different word for this meaning. What is it? (You can probably figure this out from the context.)

1.1 Conics from sum or difference of distances to two points (10 points)

Answer Questions 1, 2, 3 on the Lab 1 sheet. First describe the traces in conventional (non-Sketchpad) mathematical language as loci of points that satisfy some distance relationship. Then give a careful answer and good explanation for Question 3.

Hint: If circles with centers F1 and F2 intersect at point Q, then QF1F2 is a triangle, so the side lengths satisfy the triangle inequality.

1.2 A locus defined by the ratio of distances to two points (5 points)

Answer Questions 4 and 5 on the Lab 1 sheet.

1.3 Symmetries of conics (5 points)

  1. Take the definition of the loci in 1.1 above. Just using these definitions, without yet knowing the coordinate equation of the loci or the fact that they turn out to be ellipses and hyperbolas, tell what are the lines of symmetry of these loci and give a good geometric explanation of why these lines are lines of symmetry.
  2. Using what you know about composition of line reflections, what does the presence of these lines of symmetry imply about the existence of a point of symmetry?

1.4 Tangent to Parabola (15 points)

This problem refers to the parabola construction in Investigation 1, Exploration 6.4 of GTC, pp. 89-90. Construct this Sketch if you have not done so. We use the same notation. 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 in the following way. Prove that if Q is any point on m different from P, then the distance from Q to line BC is less than the distance from Q to A, so Q is not on the locus (this means m intersects the locus in one point).
  3. Set up a coordinate system in which line BC is the y-axis and A is point (a,0) on the x-axis. Derive the equation satisfied by the points P and show that this agrees with the usual equation of a parabola.
  4. If point P = (p1, p2) is on the parabola, what is the equation of m for this P?

1.5 Hyperbolas from circles (10 points)

This problem refers to the ellipse/hyperbola construction in Investigation 2, Exploration 6.4 of GTC, pp. 91-92. Construct this Sketch if you have not already done so.

  1. 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.
  2. 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?