# Guessing the shape of curves

1. Jan 10, 2012

### urdsirdusrnam

This is from Spivak's Calculus.
In an appendix, he defines polar coordinates. One of the exercises in this appendix is showing that the lemniscate, whose polar equation is:
r^2=2(a^2)*cos(2theta)
is the set of points P that satisfy that the product of the distances from said point to two fixed points (-a,0) and (a,0) is "a" squared. This is an excercise from that appendix:
Make a guess about the shape of the curves formed by the set of all points P that satisfying d_1*d_2=b, when b>a^2 and when b<a^2.
I'm helpless at this part. I've shown that the curves will be symmetrical with the origin as center of symnmetry and that the first one intersects both the x and y axes twice each while the second one intersects the x-axis four times whithout intersecting the y-axis at all.
Is there any easy way of picturing these curves that's been eluding me?
I apologise for my Latex iliteracy.

2. Jan 11, 2012

### Stephen Tashi

Did Spivak say what d_1 and d_2 were?

3. Jan 16, 2012

### urdsirdusrnam

Yes d1 was the distance from the point (-a,0) to a point in the curve P(x,y) and d2 is the distance from the point (a,0) to the same point.

I could graph the curves only because I typed the equations on Wolfram. Is there any algebraic/geometric argument I could use to graph them without a plotter?