## 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 1. Write down an equation relating the and components of all points on 2. Replace by and by (Note that )

3. 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. 