### 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?

- We want the slope of the line from P to Q to be equal to m.
- We want the slope of the line from P to Q to be equal to n.
- We want P to be on the left parabola (some condition relating a
and b?).
- We want Q to be on the right parabola (some condition relating c
and d?).
- We want m to be the slope of a line tangent to the left parabola
at the point P.
- 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.