This is from Spivak's Calculus.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Guessing the shape of curves

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