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urdsirdusrnam
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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.
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.