Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

For what values of \(B\) is the graph of \(x^{2}+B x y+y^{2}=1\) (a) an ellipse (b) a circle (c) a hyperbola (d) two parallel lines

Step-by-Step Solution

Answer

(a) \(-2 < B < 2\), (b) \(B = 0\), (c) \(B > 2\) or \(B < -2\), (d) \(B = 2\) or \(B = -2\)

Step 1: Identify Conic Sections Requirements

The given equation is of the form \(Ax^2 + Bxy + Cy^2 = 1\). To determine the type of conic section, we need to analyze the discriminant of the quadratic, defined for conics as \(D = B^2 - 4AC\). Given that \(A = 1\), \(B = B\), and \(C = 1\), the discriminant \(D\) simplifies to \(D = B^2 - 4\).

Step 2: Determine Conditions for an Ellipse

An ellipse is formed when the discriminant \(D < 0\). For our equation, this means we need \(B^2 - 4 < 0\). Solving this inequality: \(B^2 < 4\), which results in \(-2 < B < 2\).

Step 3: Determine Conditions for a Circle

A circle is a special case of an ellipse where \(B = 0\). Thus, for the equation \(x^2 + By + y^2 = 1\) to represent a circle, \(B\) needs to be exactly \(0\).

Step 4: Determine Conditions for a Hyperbola

A hyperbola occurs when the discriminant \(D > 0\). For our equation, this requires \(B^2 - 4 > 0\), which simplifies to \(B^2 > 4\). Solving gives \(B > 2\) or \(B < -2\).

Step 5: Determine Conditions for Two Parallel Lines

The conic becomes degenerate (such as two parallel lines) when \(D = 0\). For the equation, this requires \(B^2 - 4 = 0\), which simplifies to \(B^2 = 4\). Solving gives \(B = 2\) or \(B = -2\).

Key Concepts

Ellipse Conditions

For a quadratic equation in the form of \[ Ax^2 + Bxy + Cy^2 = 1, \] to be represented as an ellipse, the discriminant must be less than zero. The discriminant for conic sections is given by the formula \[ D = B^2 - 4AC. \] In the given exercise, where \( A = 1 \), \( B = B \), and \( C = 1 \), we simplify the discriminant to \[ D = B^2 - 4. \] For an ellipse to exist, we need:

• \( D < 0 \)
• \( B^2 - 4 < 0 \)
• \( B^2 < 4 \)

This inequality deduces to \( -2 < B < 2 \). This means that for any value of \( B \) within this range, the equation will describe an ellipse inhabited with all the properties of such a shape such as having a major and minor axis.

Circle Identification

Circles represent a special instance of ellipses. In the case of a circle, the coefficient \( B \) that links \( x \) and \( y \) together must be zero. In the equation form \[ x^2 + Bxy + y^2 = 1, \] the circle appears when \( B = 0 \). When \( B \) is exactly zero, the equation simplifies to the standard circle form \[ x^2 + y^2 = 1. \] This represents a circle centered at the origin, with a radius of 1. Thus, when asked to identify the values of \( B \) for which the conic section is a circle, remember it’s the unique case where \( B = 0 \). This configuration ensures all possible angles of intersection are equal, establishing the perfect roundness characteristic of circles.

Hyperbola Conditions

A hyperbola is identified when the discriminant is greater than zero. Using the formula \[ D = B^2 - 4, \] we derive the condition for a hyperbola as \[ B^2 - 4 > 0. \] Solving this inequality gives us:

• \( B^2 > 4 \)
• \( B > 2 \) or \( B < -2 \)

Thus, the values of \( B \) that result in a hyperbola are those that lie outside the interval \( -2 \) and \( 2 \). These values reveal an open-ended nature in the curve, leading to two distinct branches of the hyperbola that approach asymptotes but never actually meet at infinity.

Parallel Lines Criteria

Parallel lines arise from degenerate cases of conics where the discriminant precisely equals zero. This likens to the case when two lines run alongside each other without intersecting. For the conic equation \[ x^2 + Bxy + y^2 = 1, \] to break down into two parallel lines, the condition requires that \[ D = B^2 - 4 = 0. \] Solving this equation leads to \( B^2 = 4 \). Thus, the possible solutions are:

• \( B = 2 \)
• \( B = -2 \)

Either of these values for \( B \) simplifies the equation into the form describing two parallel lines. It's a situation where the discriminant balance creates a degenerate conic, resulting in non-curved, straightforward paths we recognize as parallel lines.