A hint for problem 1 of workshop set #1

Most of a message to me
... my friend ... and I have been working on the first homework question literally for hours, and havent gotten very far, except I've written a few equations taking up an entire page that have gotten us literally nowhere. ... Is there anything you can tell us to get us on the right track? Thanks!

My answer
Errr ... it should not be THAT hard. The most important thing, and I am serious, is to realize that there are many, many letters in the alphabet. You should NOT use x and y to mean more than one quantity, and it would be very, very good to use other letters so you know what you are talking about.

So I might begin in the following way: Suppose P is a point on the left parabola (opening up), and P has coordinates (a,b). Suppose Q is a point on the right-hand parabola, opening down, and Q has coordinates (c,d). Suppose the slope of the left parabola at P is m and the slope of the right parabola at Q is n. Lots of letters: a and b and c and d and m and n.

Now, what do we want?

  1. We want the slope of the line from P to Q to be equal to m.
  2. We want the slope of the line from P to Q to be equal to n.
  3. We want P to be on the left parabola (some condition relating a and b?).
  4. We want Q to be on the right parabola (some condition relating c and d?).
  5. We want m to be the slope of a line tangent to the left parabola at the point P.
  6. We want n to be the slope of a line tangent to the right parabola at the point Q.
Write out everything in terms of these variables. Do NOT try to re-use x and y or any such shortcuts. There are 6 variables and there are 6 equations. Hey, most of the equations will be rather simple and solving this "system" of equations should not be that difficult.

Let me know if this helps, please.

Thanks for writing.

Maintained by greenfie@math.rutgers.edu and last modified 9/4/2005.