Problem 1 - Analytical Geometry: 2D and 3D

Question

If \(P S P^{\prime}\) and \(Q S Q^{\prime}\) are two focal chords of a conic cutting each other at right angles then prove that \(\frac{1}{S P . S P^{\prime}}+\frac{1}{S Q \cdot S Q^{\prime}}=\) a constant.

Step-by-Step Solution

Answer

Based on the given information, we can prove that for any conic and any two pairs of focal chords intersecting at right angles, the sum of the reciprocals of the lengths of the chord products is a constant. We achieve this by finding the slopes of the lines, applying properties related to focal chords of conics, and then writing an expression for the sum of reciprocals. The final expression we get is \(\frac{k_1 + k_2}{k_1 k_2} = \text{constant}\), which shows that the sum of reciprocals of the lengths is a constant.

Step 1: Start with given information

We are given that \(P S P^{\prime}\) and \(Q S Q^{\prime}\) are two focal chords of a conic. Let \(S\) be the focus, and let \(P\) and \(P^{\prime}\) be on the conic such that the chords \(PS\) and \(SP^{\prime}\) intersect at a right angle. Similarly, let \(Q\) and \(Q^{\prime}\) be on the conic such that \(S Q\) and \(SQ^{\prime}\) intersect at a right angle.

Step 2: Find the slopes of the lines

We know that the product of the slopes of two perpendicular lines is -1. First, we need to find the slopes of lines \(PS\), \(SP^{\prime}\), \(SQ\), and \(SQ^{\prime}\). We will denote the coordinates of point \(P\) as \((x_1, y_1)\), point \(P^{\prime}\) as \((x_2, y_2)\), point \(Q\) as \((x_3, y_3)\), and point \(Q^{\prime}\) as \((x_4, y_4)\). Let the slope of the line \(PS\) be \(m_1\), and the slope of the line \(SP^{\prime}\) be \(m_2\). Similarly, let the slope of the line \(SQ\) be \(m_3\), and the slope of the line \(SQ^{\prime}\) be \(m_4\). We then have four equations: $$m_1 = \frac{y_1 - y_S}{x_1 - x_S}$$ $$m_2 = \frac{y_2 - y_S}{x_2 - x_S}$$ $$m_3 = \frac{y_3 - y_S}{x_3 - x_S}$$ $$m_4 = \frac{y_4 - y_S}{x_4 - x_S}$$ Since lines \(PS\) and \(SP^{\prime}\) are perpendicular, it must be true that the product of their slopes is -1: $$m_1 m_2 = (x_1 - x_S)(y_2 - y_S) - (y_1 - y_S)(x_2 - x_S) = -1$$ Similarly, since lines \(SQ\) and \(SQ^{\prime}\) are perpendicular, it must be true that the product of their slopes is -1: $$m_3 m_4 = (x_3 - x_S)(y_4 - y_S) - (y_3 - y_S)(x_4 - x_S) = -1$$

Step 3: Apply properties related to focal chords of conics

We know that the distance between a point and its corresponding point on a conic is a constant. Therefore, we can write: $$SP \cdot SP^{\prime} = (x_1 - x_S)(x_2 - x_S) + (y_1 - y_S)(y_2 - y_S) = k_1$$ $$SQ \cdot SQ^{\prime} = (x_3 - x_S)(x_4 - x_S) + (y_3 - y_S)(y_4 - y_S) = k_2$$ where \(k_1\) and \(k_2\) are constants.

Step 4: Write an expression for the sum of reciprocals

Since our goal is to relate the inverse of the lengths of the given chord products, we can write: $$\frac{1}{SP \cdot SP^{\prime}} + \frac{1}{SQ \cdot SQ^{\prime}} = \frac{1}{k_1} + \frac{1}{k_2} = \frac{k_1 + k_2}{k_1 k_2} = \text{constant}$$ So, for any given conic and any two pairs of focal chords intersecting at right angles, the sum of the reciprocals of the lengths of the chord products is a constant.

Key Concepts

Analytical Geometry

At the heart of analytical geometry, also known as coordinate geometry, is the use of a coordinate system to define and analyze geometric figures. Through the lens of Cartesian coordinates, we can easily describe the location and algebraic relationships of points, lines, and shapes on a plane.

The powerful bond between algebra and geometry in this discipline allows us to solve complex geometrical problems that would otherwise be cumbersome with classical methods. For instance, the concept of a conic section can be embedded in equations and studied by their qualities — such as axis, directrix, and focus. When dealing with problems like those involving focal chords, analytical geometry equips us with methods to find slopes, distances, and the nature of the relationship between algebraic variables to geometric features.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include parabolas, ellipses (which encompass circles), and hyperbolas. Each of these shapes has their own distinct set of properties and equations.

Focal chords are line segments within conic sections that pass through a focus of the conic. The focus is a special point that, in combination with the directrix, defines the shape of a conic. A fascinating trait of these chords is that properties of their lengths or interrelationships can often be expressed as constants, emphasizing the symmetrical nature of conics. The exercise in question showcases this by demonstrating that the sum of reciprocals of the products of the lengths of focal chords intersecting at right angles is a constant value, which is deeply rooted in the inherent properties of conic sections.

Slope of a Line

The slope of a line in the coordinate plane is a measure of its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In simpler terms, it's how much the 'y' value changes for a one unit increase in the 'x' value.

An essential characteristic is that the slopes of perpendicular lines are negative reciprocals of each other. This condition is fundamental when examining focal chords of conics that intersect at right angles, as it allows us to establish relationships between their slopes — denoted as \( m_1, m_2, m_3, \text{and} m_4 \) in the provided solution — and utilize these relationships to prove the constancy of their corresponding lengths' reciprocal sums.