Question

1. Find a Cartesian equation for the curve and identify it: r²sin2θ = 1.

Answer 1

polar equation of curve is given in the question, $r^2sin2\theta=1$

we know, sin2x = 2sinx.cosx
so, $sin2\theta=2sin\theta cos\theta$

now, $r^2(2sin\theta vos\theta)=1$

or, $(rsin\theta)(rcos\theta)=1$

we know, relation between Cartesian and polar system, $x=rcos\theta,y=rsin\theta$
where, $\theta$ is angle made by r with x axis.

so, $(rsin\theta)(rcos\theta)=yx=1$

hence, Cartesian equation is xy = 1

do you not think , xy = 1 is similar like xy =c²
yes of course , it is similar. so xy = 1 is hyperbolic equation as shown in figure.

Answer 2

Solution:

Given curve is

r²sin2θ = 1

Use the following substitutions to convert Cartesian coordinates

x=r cosθ ......(1)

y=r sinθ ......(2)

Now,

r²sin2θ = 1

r².2.sinθ cosθ = 1

r².2.$(\frac{y}{r})(\frac{x}{r})$ = 1

r².2.$(\frac{xy}{r^2})$ = 1

2xy=1

xy=1/2

This equation is of the form $xy=c^2$

therefore the given polar equation represents a rectangular hyperbola