Finding the Equation of the Image of a Curve Under the Reciprocal Function
To find the equation of the imageof a pathunder the reciprocal function

Write down an equation relating theandcomponents of all pointson

Replacebyandby(Note that)

Simplify the resulting equation to obtain an equation relating theandcomponents of all pointson the image
Example: Find the equation of the image of the circleunder the reciprocal function.
The centre of the circle is atand the radius of the circle is 3. The cartesian equation of the circle is
Making the substitution in 2 gives
Expanding gives
Collecting the first and fourth terms gives
Cancelling and clearing fractions gives
Hence
Completing the square gives
This is the equation of a circle centrewith radius
In general the reciprocal function sends:
a line through the origin to a line through the origin
a line not through the origin to a circle through the origin
a circle through the origin to a line not through the origin
a circle not through the origin to a circle not through the origin.