Problem 21 - A History of Mathematics: An Introduction

Question

Show that if the point \(p^{\prime}\) lies on the polar \(\pi\) of a point \(p\) with respect to a conic \(C\), then \(\pi^{\prime}\), the polar of \(p^{\prime}\), goes through \(p\). (Hint: Assume first that \(C\) is a circle.)

Step-by-Step Solution

Answer

Question: Show that if a point \(p'\) lies on the polar \(\pi\) of a point \(p\) with respect to a conic \(C\), then the polar \(\pi'\) of point \(p'\) goes through point \(p\). Answer: We can prove this by first identifying two lines tangent to the conic at points \(p\) and \(p'\), and finding their intersection points with the conic. Upon proving that \(p\), \(C\), and \(D\) are collinear, we can show that they lie on the polar of \(p'\) with respect to the conic, which means that the polar \(\pi'\) of point \(p'\) goes through point \(p\).

Step 1: Identify two lines tangent to the conic at \(p\) and \(p'\)

Let's consider two lines \(t_1\) and \(t_2\) that are tangent to the conic \(C\) at points \(p\) and \(p'\), respectively.

Step 2: Find the intersection points of \(t_1\) and \(t_2\) with the conic

Now, let's find the intersection points \(A\) and \(B\) of line \(t_1\) with \(C\). Similarly, find the intersection points \(C\) and \(D\) of line \(t_2\) with \(C\).

Step 3: Prove that \(p\), \(C\), and \(D\) are collinear

Since \(p'\) lies on the polar of \(p\) with respect to \(C\), we know that the cross ratio (notation \(\mathrm{cr}\)) \(\mathrm{cr}(A,B,p',p)=p\cdot A\cdot p'\cdot B = -1\). By definition, \(\pi'\) is the polar of \(p'\) with respect to \(C\). This means that the cross ratio \(\mathrm{cr}(A,B,p',C) = p'\cdot A\cdot p\cdot C = -1\) and \(\mathrm{cr}(A,B,p',D) = p'\cdot A\cdot p\cdot D = -1\). Since both cross ratios are equal to -1, \(p\), \(C\), and \(D\) are collinear.

Step 4: Show that \(p\), \(C\), and \(D\) lie on the polar of \(p'\) with respect to the conic

Since \(p\), \(C\), and \(D\) are collinear, they must lie on a line. And because \(p\), \(C\), and \(D\) lie on the line that satisfies the property of the polar of \(p'\) with respect to the conic (\(\mathrm{cr}(A,B,p',p)=\mathrm{cr}(A,B,p',C)=\mathrm{cr}(A,B,p',D)=-1\)), this line must be the polar \(\pi'\) of \(p'\). Hence, we have shown that the polar \(\pi'\) of \(p'\) goes through point \(p\).

Key Concepts

Polar of a Point

In geometry, particularly in the study of conic sections, the concept of a polar related to a point is fundamental. When you have a point \( p \) with respect to a conic, such as a circle, ellipse, parabola, or hyperbola, the polar is a unique line associated with that point. Here's how it works:

• The polar of a point, denoted often as \( \pi \), is the locus of all points that have a particular relationship with the point and the conic. This relationship involves tangent lines drawn from the point to the conic.


• From any point outside the conic, you can draw two tangent lines that just "kiss" the conic. The points where these tangents touch the conic determine the polar.


• The fascinating property of the polar is its symmetry. If a point \( p' \) lies on the polar \( \pi \) of the point \( p \), then conversely, the polar of \( p' \) must pass through the point \( p \).


• This symmetric relationship can be demonstrated using cross ratios, which show a consistent geometric relationship.

Understanding polars helps in solving various problems related to tangents and secants in conic sections.

Tangent Lines

Tangent lines in geometry are crucial when studying curves and conics. A tangent line to a curve at a given point is a straight line that just touches the curve at that point. For conic sections, tangent lines have special properties:

• For a circle, the tangent at any point on the circle is perpendicular to the radius drawn to the point of tangency.


• In the case of an ellipse, parabola, or hyperbola, the tangent at a point has a specific slope determined by the derivative at that point (or fitting geometric construction).


• Tangent lines are used to define polars. If you have two tangent lines from a point on a conic, the connection forms the basis for identifying the polar line associated with that point.


• Lines that are tangent from a point can help locate intersection points, which are crucial in determining the polar and solving geometric problems involving conics.

The intersection and interplay of tangent lines are vital in understanding conic properties and solving related exercises.

Properties of Cross Ratios

Cross ratios are a powerful concept in projective geometry. They help understand and solve geometry problems involving lines and points, especially with conics.

• A cross ratio is a value that remains constant for four collinear points or four points concyclic (lying on the same circle).


• Mathematically, for four collinear points \( A, B, C, \) and \( D \), the cross ratio \( \mathrm{cr}(A, B, C, D) \) is given by \( \frac{(A-C)(B-D)}{(A-D)(B-C)} \).


• In conics, if a point lies on the polar of another point, specific cross ratios equal -1. This equality indicates a harmonic division, which is fundamental to understanding polar and tangent properties.


• The symmetry in cross ratios is what confirms the geometric relationships we discussed, like the mutual polarity of points and lines in the conic.


• They provide a coherent method to prove many relationships like collinearity or conicity in projective geometry uses.

Cross ratios thus serve as the mathematical glue in many geometric theorems involving conics.