As I said in the previous post, there is a duality:
Points on a Curve (Geometry) Solutions of an Equation (Algebra)
This means we can answer geometric questions using algebra and answer algebraic questions using geometry.
Consider the following two questions:
- Find the tangents to a circle of a given slope.
- Find the tangents to a circle through a given point.
Both can be answered using the duality principle.
Find the tangents to the circle
(a) parallel to the line
(b) through the point [caution: the numbers here are disgusting]
Solution (a) i:
First of all a sketch (and the remark that a tangent is a line):
Here we see the circle and the line on the bottom left. The two tangents we are looking for are as shown. They have the same slope as and have only one intersection with . These two pieces of information will allow us to find the equations of the tangents.