- #1
jamesbob
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The lemniscate of Bernoulli is the curve that is the locus of points the product of whose distances from two fixed centres (called the foci) a distance of 2c apart is the cosntant [tex][c^2[/tex]. If the foci have Cartesian coordinates [tex](\pmc, 0)[/tex] the Cartesian equation of the lemniscate is
or
a) Show that the lemniscate of Bernoulli may be expressed parametrically by
where [tex] t \epsilon[-\pi, \pi). [/tex] For t out of this interval the curve repeats on itself.)
[tex]([x-c]^2 + y^2)([x + c]^2 + y^2) = c^4[/tex]
or
[tex](x^2 + y^2)^2 = 2c^2(x^2 - y^2).[/tex]
a) Show that the lemniscate of Bernoulli may be expressed parametrically by
[tex] x(t) = \sqrt{2c}\frac{\cost}{1 + \sin^2t}[/tex], [tex] y(t) = \sqrt{2c}\frac{costsint}{1 + sin^2t}[/tex]
where [tex] t \epsilon[-\pi, \pi). [/tex] For t out of this interval the curve repeats on itself.)