Expert: Paul Klarreich - 5/4/2006

Question
Sorry, I made a mistake. This is the correct equation:

x^4 = 4(x^2-y^2)
1) Find the four points where the tangent is horizontal
2) Find the two points where the tangent is vertical
Thank you very much
Hope to hear from you soon
Emma  

Answer
Hi, Emma,
An Eight Curve (or Gerono Lemniscate): Question revised
Question:  Sorry, I made a mistake. This is the correct equation:

x^4 = 4(x^2-y^2)
1) Find the four points where the tangent is horizontal
2) Find the two points where the tangent is vertical
Thank you very much
Hope to hear from you soon
Emma  
-----------------------------
That's better.  Try implicit differentiation:

x^4 = 4x^2 - 4y^2

4x^3 = 8x - 8y y'

8y y' = 8x - 4x^3

2y y' = 2x - x^3
    2x - x^3
y' = ---------
       2y
Now you are in business, sort of.  You will have a tangent that is:

A. horizontal when y' = 0, which means the top is zero.

2x - x^3 = 0
x(2 - x^2) = 0

x = 0  or  x = +- sqrt(2)

Those are the x-coordinates.  We want y as well, for each.  Solve in the original equation:

x = 0:  0 = 0 - 4y^2 -->  y = 0.  That is no good, because y' will be undefined there.

x = sqrt(2):  4 = 4(2) - 4y^2 --> 1 = 2 - y^2  --> y^2 = 1  --> y = +- 1.

So the four points are  (+-sqrt(2), +-1)

B. Vertical when y' is undefined, which means y = 0.  Now solve again:

y = 0:  x^4 = 4x^2  -->  x^2 = 4 -->  x = +- 2.

So the two points are:  (+-2, 0)

You could use your graphing calculator (sorry, I don't have one, so I can't tell you how) to confirm this.  Plot  y = + (1/2)sqrt(4x^2 - x^4) and minus that as separate graphs.

Just remember this 'trick' -- a fraction is zero when the top is zero, and undefined when the bottom is zero.  If both are, then all bets are off.