Finding the Equation of the Image of a Curve Under the Reciprocal Function
To find the equation of the image
of a path
under the reciprocal function
Write down an equation relating the
and
components of all points
on 
Replace
by
andby
(Note that
)Simplify the resulting equation to obtain an equation relating the
and
components of all points
on the image
Example: Find the equation of the image of the circle
under the reciprocal function.
The centre of the circle is at
and 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 centre
with 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.
