To find a polar equation corresponding to the Cartesian equation xy = 1, we start by recalling the relationships between Cartesian and polar coordinates. In polar coordinates, x and y can be expressed as:
x = r imes ext{cos}( heta)y = r imes ext{sin}( heta)
Here, r is the radial distance from the origin, and θ is the angle measured from the positive x-axis.
Substituting these expressions for x and y into the equation gives:
(r imes ext{cos}( heta))(r imes ext{sin}( heta)) = 1
This simplifies to:
r^2 imes ext{cos}( heta) imes ext{sin}( heta) = 1
Next, we can use the identity 2 imes ext{cos}( heta) imes ext{sin}( heta) = ext{sin}(2 heta) to rewrite the equation:
r^2 imes rac{1}{2} imes ext{sin}(2 heta) = 1
This leads to:
r^2 = rac{2}{ ext{sin}(2 heta)}
To express r in terms of θ, we take the square root:
r = rac{ ext{sqrt}(2)}{ ext{sqrt}( ext{sin}(2 heta))}
Thus, the polar equation corresponding to the Cartesian equation xy = 1 is:
r = rac{ ext{sqrt}(2)}{ ext{sqrt}( ext{sin}(2 heta))}