I'm trying to attack a question on folium of Descartes from J.Stewart Calculus book.(adsbygoogle = window.adsbygoogle || []).push({});

Here are the ones I've been trying so far and got stuck:

btw parametric equations are:

x=[tex]\frac{3t}{1+t^{3}}[/tex]

y=[tex]\frac{3t^{2}}{1+t^{3}}[/tex]

(a) Show that if (a, b) lies on the curve, then so does (b, a); that is, the curve is symmetric

with respect to the line y=x...

(c) Show that the line y=-x - 1 is a slant asymptote.

(e) Show that a Cartesian equation of this curve is x[tex]^{3}[/tex] + y[tex]^{3}[/tex] = 3xy.

A:

I suppose that if (a, b) is on the curve, then we should find a parameter t such that it

solves the following system:

[tex]\frac{3t}{1+t^{3}}[/tex] = b

[tex]\frac{3t^{2}}{1+t^{3}}[/tex] = a

Adding these up then expanding (1 + t[tex]^{3}[/tex]) and canceling

I end up with a quadratic equation in this form:

t1,2 = [tex]\frac{-(3+a+b)+/-\sqrt{(3 + a + b)^{2} - 4(a+b)^{2}}}{2(a+b)}[/tex]

I checked this up for a value of t=3 giving me some values for a and b, one of the roots is 1/3 which gives a point (b, a); But what is the meaning of the second root? I tried to constraint the discriminator but it does not make much sense to impose limitations on value a+b, cause they should lie on the curve, but that does not mean that for every point(a,b) the equation will have a solution.

Am I doing something wrong?

C:

Completely lost here; I tried looking at the limits of both parametric equations as t goes to -[tex]\infty[/tex] and +[tex]\infty[/tex], but they approach 0 from either left or right, which can be right...Thou I'd expect both limits to be [tex]\infty[/tex] as a point approaches the asymptote, what values of t should I look at then; or would it be right look at the ration of both limits? e.g how faster y increases compared to x...

E:

Similar story; From what I've seen before, the way to solve is to eliminate the parameter from one of the equations... I've tried different algebraic manipulations with no success, probably there's another approach or some "trick" :)

I'm trying to crack this task for 2 nights already and it start to become frustrating that I can't really solve most of the questions in it/\

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# Folium of Descartes

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