## 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.