Problem 57 - Algebra 2

Question

Consider equations of the form \(A x^{2}+B y^{2}+C x+D y+E=0\) a. What must be true about \(A\) and \(B\) for the graph of the equation to be a circle? To be an ellipse? To be a hyperbola? To be a parabola? b. Suppose \(A=1\) and \(B=1 .\) Must the graph be a circle? Explain. c. Suppose \(A=1, B=-1,\) and \(C=D=E=0 .\) Describe the graph.

Step-by-Step Solution

Answer

a. Circle: \(A\) and \(B\) must be equal and positive. Ellipse: \(A\) and \(B\) are positive but not necessarily equal. Hyperbola: \(A\) and \(B\) should have opposite signs. Parabola: Either \(A\) or \(B\) should be zero. b. Yes, the graph will indeed be a circle. c. The graph will be a hyperbola centered at the origin (0,0).

Step 1: Identify Properties of Graphs

Here is the general information about quadratic forms in two variables:\nA. For the graph to be a circle, the coefficients \(A\) and \(B\) must be equal and positive.\nB. For an ellipse, \(A\) and \(B\) are positive but not necessarily equal.\nC. For a hyperbola, \(A\) and \(B\) should have opposite signs.\nD. And for a parabola, either \(A\) or \(B\) should be zero.

Step 2: Examine Conditions for a Circle

Given \(A=1\) and \(B=1\), it satisfies the condition for a circle, with \(A = B\) and both are positive.

Step 3: Describe the Hyperbola Graph

Given \(A=1\), \(B=-1\) and \(C = D = E = 0\). Since \(A\) and \(B\) are opposite in sign and C, D, E are zeros, this forms the equation \(x^{2} - y^{2} = 0\), which represents a hyperbola centered at the origin (0,0).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in understanding conic sections and are typically expressed in the general form \[ Ax^2 + By^2 + Cx + Dy + E = 0. \] These equations describe a curve on a plane. The properties of this curve are determined by the values of the coefficients \(A\), \(B\), \(C\), \(D\), and \(E\).

• If both \(A\) and \(B\) are positive, the curve can represent a circle or an ellipse, depending on whether they are equal or not.
• If \(A\) and \(B\) have opposite signs, the curve forms a hyperbola.
• If either \(A\) or \(B\) is zero, it simplifies to a parabola.

Understanding the relationship between these coefficients is crucial for identifying the type of conic section represented by the equation. Each type has its unique geometric properties, which can be visualized by plotting the curve defined by the quadratic equation.

Circles

A circle is a perfect geometric shape where all points on the boundary are equidistant from the center. To express a circle algebraically in terms of a quadratic equation, the requirement is that the coefficients \(A\) and \(B\) should be equal and positive, i.e., \(A = B \). Such an equation takes the form \[ Ax^2 + By^2 + Cx + Dy + E = 0, \] where \(A = B\), leading to the simplified formula \[ (x - h)^2 + (y - k)^2 = r^2, \] with \((h, k)\) as the center and \(r\) as the radius. This formula emphasizes the equal distribution of points from the center, a defining property of a circle. In the exercise, given \(A = 1\) and \(B = 1\), it satisfies the circle condition.

Ellipses

An ellipse resembles a stretched circle and is defined by a quadratic equation where \(A\) and \(B\) are positive but not equal. This inequality creates the elongated shape of the ellipse. The general form becomes \[ \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1, \]where \(a\) and \(b\) are the distances from the center to the covertex and focus, respectively.

• If \(a > b\), the ellipse is stretched more in the x-direction.
• If \(b > a\), it stretches more in the y-direction.

This variation from a circle to an ellipse reflects the variability of the coefficients \(A\) and \(B\). Ellipses naturally appear in planetary orbits, making this concept significant in astronomy.

Hyperbolas

A hyperbola consists of two separate curves called branches, which are related but never meet. This is represented in a quadratic equation when \(A\) and \(B\) have opposite signs, leading to the form \[ Ax^2 - By^2 = C. \] The standard form is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \]where \(a\) and \(b\) determine the distance to the vertices and the asymptotes' steepness.

• This equation implies the curves approach but never touch the asymptotes.
• The center, vertices, and foci are critical points in understanding a hyperbola's shape.

In the solution provided, with \(A = 1\), \(B = -1\), and \(C = D = E = 0\), the equation simplifies to \[ x^2 - y^2 = 0, \] indicating a hyperbola opening along the x and y axes, effectively graphed as a pair of intersecting lines through the origin.