How Do Polar Coordinates Reveal the Shape of Curves in Spivak's Calculus?

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Polar coordinates in Spivak's Calculus reveal the shape of curves by defining relationships between distances to fixed points. The lemniscate, represented by the equation r^2=2(a^2)*cos(2theta), illustrates this concept through its symmetry and intersections with axes. The discussion explores the curves formed by points satisfying the product of distances d_1 and d_2, with varying conditions for b relative to a^2. The curves exhibit specific symmetry and intersection behaviors, prompting questions about visualizing them without computational tools. Overall, the challenge lies in understanding the geometric properties of these curves based on their polar equations.
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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.
Thanks in advance.
 
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Did Spivak say what d_1 and d_2 were?
 
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?
 
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