Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

For an ellipse, let \(p\) and \(q\) be the distances from a focus to the two vertices. Show that \(b=\sqrt{p q}\), with \(2 b\) being the minor diameter.

Step-by-Step Solution

Answer

The minor semi-diameter is given by the relation \\(b = \\sqrt{pq}\\).

Step 1: Review Definitions

An ellipse has two foci and two vertices. The vertices lie on the major axis of the ellipse. Assuming the major axis is horizontal, we define the center of the ellipse at the origin \(0, 0\), and the vertices at \((a, 0)\) and \((-a, 0)\)\, where \(a \geq b\) is the semi-major axis length. The foci are at \((c, 0)\) and \((-c, 0)\)\, where \(c \leq a\).

Step 2: Use the Definition of an Ellipse

By definition of an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equals the length of the major axis. In standard form for a point on the ellipse \((x, y)\): \sqrt{(x - c)^2 + y^2} + \sqrt{(x+c)^2 + y^2} = 2a\.

Step 3: Express distances p and q

Consider the point on the ellipse that is a vertex. Distance \(p\) from a focus to \((a, 0)\) is \(|a-c|\) and distance \(q\) from the other focus to the same vertex is \(|a+c|\). This gives us \p = a-c\ and \q = a+c\.

Step 4: Derive the relation for b

We know that \(b\) is related to \(a\) and \(c\) by \(b^2=a^2-c^2\). From \(p = a-c\) and \(q = a+c\), calculate \(pq = (a-c)(a+c) = a^2 - c^2\). Thus, \(b^2 = pq\), and it follows that \(b = \sqrt{pq}\).

Key Concepts

Ellipse Definitions

An ellipse is a shape defined by a collection of points. For every point on the ellipse, the sum of the distances to two fixed points, known as foci, is constant. This geometric property distinguishes ellipses from other shapes, such as circles or parabolas.

For visualization, imagine a stretched circle. The longest diameter of this "stretched circle" is called the major axis, and the shortest is the minor axis. The points where the major axis intersects the ellipse are the vertices. In a standardized form, we assume the ellipse's center is at the origin \(0, 0\), with the vertices along the x-axis at positions \((a, 0)\) and \((-a, 0)\), where \(a\) is the semi-major axis length.

Meanwhile, the foci are positioned along the major axis as well, but located closer to the center at \((c, 0)\) and \((-c, 0)\) with \(c < a\). This arrangement helps maintain the ellipse's characteristic of equal total distances from any point on the ellipse to the foci.

Ellipse Equation

The standard equation of an ellipse helps describe its shape and properties mathematically. When centered at the origin and with its major axis along the horizontal, the equation is given as: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Here, \(a\) represents the semi-major axis, and \(b\) is the semi-minor axis. These values help define how "stretched" the ellipse is in each dimension. For a horizontal orientation like our example, the semi-major axis \(a\) is greater than or equal to the semi-minor axis \(b\).

This equation is derived from the geometric definition of an ellipse, specifically how each point maintains a constant sum of distances to the foci. Therefore, this relation is crucial for understanding the proportions and orientation of the ellipse in space.

Ellipse Foci and Vertices

Foci and vertices are two fundamental components of an ellipse's geometry. Understanding their positioning is key to grasping ellipse properties.

The foci are two important points located along the major axis. They are positioned symmetrically about the center of the ellipse at coordinates \((c, 0)\) and \((-c, 0)\). The distance from the center to each focus is denoted by \(c\), and it influences the "shape" of the ellipse.

The vertices, on the other hand, are the outermost points along the major axis, found at \((a, 0)\) and \((-a, 0)\). These points indicate the extent of the ellipse's reach along its longest diameter. Together, they play a crucial role in defining the equation of an ellipse and the interconnected relationship between the semi-major and semi-minor axes, described by \(b^2 = a^2 - c^2\).

Mathematical Proofs

Mathematical proofs around geometrical shapes like ellipses often involve demonstrating relationships between different elements of the figures. In the context of this exercise, the goal is to show the relationship involving the distances \(p\) and \(q\) from a focus to the vertices, and how they connect to the semi-minor axis \(b\).

From the center \((0, 0)\), the vertex \((a, 0)\) is approached by two distances: \(p = a-c\) from one focus and \(q = a+c\) from the other. Understanding these distances is key to deriving the relationship \(b^2 = a^2-c^2\). Substituting \(p\) and \(q\) into this equation results in computing \(pq = a^2 - c^2\), which confirms \(b = \sqrt{pq}\).

This proof demonstrates how algebraic manipulation and understanding geometric definitions can reveal true but sometimes hidden relationships within a shape.