Problem 66 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that, if \(A+C\) and \(\Delta=4 A C-B^{2}\) are both positive, then the graph of \(A x^{2}+B x y+C y^{2}=1\) is an ellipse (or circle) with area \(2 \pi / \sqrt{\Delta}\). (Recall from Problem 55 of Section \(10.2\) that the area of the ellipse \(x^{2} / p^{2}+y^{2} / q^{2}=1\) is \(\left.\pi p q .\right)\)

Step-by-Step Solution

Answer

If \(A + C > 0\) and \(\Delta > 0\), the given conic is an ellipse with area \(2\pi/\sqrt{\Delta}\).

Step 1: Understand the Exercise

To confirm that a given equation represents an ellipse, specific conditions must be satisfied. Specifically, certain relationships are linked to the coefficients of the equation, and we have some known formulas for calculating ellipse properties.

Step 2: Identify the General Form

The given equation is of the form \(A x^{2} + B x y + C y^{2} = 1\). We need to identify if it can represent an ellipse when \(A + C > 0\) and \(\Delta = 4AC - B^2 > 0\).

Step 3: Use Discriminant Condition for Ellipses

For the conic section given by the equation \(Ax^2 + Bxy + Cy^2 = 1\), it represents an ellipse if the discriminant \(\Delta = 4AC - B^2\) is positive.

Step 4: Verify the Conditions

Check the conditions: \(A + C > 0\) guarantees the leading quadratic terms can depict a closed shape, while \(\Delta = 4AC - B^2 > 0\) confirms the conic is an ellipse.

Step 5: Express in Standard Ellipse Form

The given conic can be transformed into \(x^2/P^2 + y^2/Q^2 = 1\) by diagonalizing the quadratic form. The standard form helps to state dimensions \(P\) and \(Q\) related to the axis lengths of the ellipse.

Step 6: Relate \\Delta to Ellipse Axes

From properties of conics, particularly ellipses, it has been established that the area of the ellipse is \(\pi p q\), where \(p\) and \(q\) are semi-axis lengths. It follows that \(pq = 1/\sqrt{\Delta}\) because of the equation form \(\frac{x^2}{P^2} + \frac{y^2}{Q^2} = 1\), leading to an area of \(\pi/\sqrt{\Delta}\).

Step 7: Compute the Area

Thus, the area of the corresponding ellipse is given by the formula: Area = \(2\pi / \sqrt{\Delta}\), as \(pq = 1/\sqrt{\Delta}\) for the chosen transformations.

Key Concepts

Conic Sections

Conic sections are a family of curves obtained by intersecting a plane with a double-napped cone. Depending on the angle of the plane with respect to the cone, the intersection can yield different shapes including circles, ellipses, parabolas, and hyperbolas.

Conic sections are defined by quadratic equations in two variables, generally represented as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Each type of conic section has a unique set of properties:

• Circle: A special case of an ellipse where the two axes are equal in length.
• Ellipse: Oval-shaped, defined when the discriminant \(\Delta = 4AC - B^2\) is positive.
• Parabola: When \(\Delta = 0\), the conic is a parabola.
• Hyperbola: Formed when \(\Delta < 0\), creating two separate curves.

Understanding these properties is crucial as they determine the nature of the conic section represented by any specific equation.

Ellipse Equation

An ellipse is characterized by its unique equation, which can take various forms. The most straightforward is its standard form: \[ \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \]

In this expression, \(p\) and \(q\) are the semi-major and semi-minor axes, respectively. They denote the distance from the center of the ellipse to its furthest and closest points along the main axes.

Ellipses also occur in quadratic form \(Ax^2 + Bxy + Cy^2 = 1\), which represents a rotated ellipse when \(B eq 0\). To simplify or relate this elliptic equation to its standard form, one often employs the transformation of axes to remove the \(xy\) term, making analysis easier by aligning with the standard orientation.

Additionally, for an ellipse, the condition \(A + C > 0\) must hold, indicating a closed curve. All these algebraic properties define the geometrical shape and orientation of the ellipse within a coordinate plane.

Area of Ellipse

The area of an ellipse is elegantly simple to calculate once you understand its properties. It is given by the formula:
\[ \text{Area} = \pi p q \]

Here, \(p\) and \(q\) are the lengths of the semi-major and semi-minor axes. This concise relationship stemming from elliptic geometry states that the area is directly proportional to the product of the two radii and the constant \(\pi\).

When the ellipse is defined in the quadratic form, the transformation can provide the axes lengths, as referenced by the condition \(pq = 1/\sqrt{\Delta}\), thus the formula becomes:

• The area formula transforms to \(2 \pi / \sqrt{\Delta}\), directly tying the area to the equation's discriminant \(\Delta\).

Understanding the area formula is crucial when dealing with ellipses derived from general conic form equations.

Quadratic Form Transformation

Quadratic form transformation is a mathematical technique used to simplify conic section equations, particularly useful for ellipses expressed in non-standard forms. The goal is to remove the cross-product term \(Bxy\) to achieve a rotational alignment of the ellipse with the coordinate axes.

This process involves diagonalizing the quadratic equation, effectively converting it into its canonical or standard form:

• Trigonometric methods or matrix operations can be applied to find the rotation angle that aligns the ellipse with the axes.
• After alignment, one can express the ellipse as \(\frac{x^2}{p^2} + \frac{y^2}{q^2} = 1\), where \(p\) and \(q\) are the transformed axis lengths.

Transformations make it easier to interpret and solve problems related to ellipses, including calculating key properties like area and intercepts. Mastering quadratic form transformation provides deeper insights into conic sections and the nature of the curves they describe.