Problem 6 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(2 x^{2}-y^{2}+4 x y-2 x+3 y=6\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the General Conic Form

A general second-degree equation in two variables can be written as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). We identify \(A, B, C, D, E, \text{ and } F\) from the given equation. Comparing with \(2x^2 - y^2 + 4xy - 2x + 3y = 6\), we have:\(A = 2\), \(B = 4\), \(C = -1\), \(D = -2\), \(E = 3\), and \(F = -6\).

Step 2: Calculate the Discriminant

The discriminant of a conic section equation is given by \(B^2 - 4AC\). Substitute \(A = 2\), \(B = 4\), and \(C = -1\) into this formula to find the discriminant:\[B^2 - 4AC = 4^2 - 4(2)(-1) = 16 + 8 = 24\].

Step 3: Determine the Conic Type

The sign of the discriminant determines the type of conic:- If \(B^2 - 4AC > 0\), the conic is a hyperbola.- If \(B^2 - 4AC = 0\), the conic is a parabola.- If \(B^2 - 4AC < 0\), the conic is an ellipse.Here, since \(B^2 - 4AC = 24 > 0\), the given equation represents a hyperbola.

Key Concepts

Discriminant

The discriminant is a critical value used to determine the type of conic section given an equation in two variables. You can recognize the discriminant formula as \(B^2 - 4AC\). This is similar to the discriminant in quadratic equations, but here it's applied to determine conics:

• If \(B^2 - 4AC > 0\), the conic section is a hyperbola.
• If \(B^2 - 4AC = 0\), the conic section is a parabola.
• If \(B^2 - 4AC < 0\), the conic section is an ellipse.

Every conic section equation can be transformed into the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Identifying the coefficients \(A\), \(B\), and \(C\) allows you to compute the discriminant and thus classify the conic section. In our example, the discriminant was found to be 24, which confirms the conic is a hyperbola.

Parabolas

Parabolas are one of the simplest conic sections. They are defined by a U-shaped course, which can open up or down, or left and right, depending on the orientation in the coordinate plane. For an equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\):- A parabola occurs only if \(B^2 - 4AC = 0\).- This situation typically suggests symmetry in one direction.In these cases, the binomial squared terms \(Ax^2\) or \(Cy^2\) generally dominate the equation, creating that single direct opening. For example, a simple quadratic like \(y = ax^2 + bx + c\) is a parabola, opening either up or down.

Ellipses

Ellipses are conic sections that resemble stretched circles. They occur when the equation in two variables with coefficients provides a certain balance:- The defining feature of an ellipse is \(B^2 - 4AC < 0\).- In an ellipse, the sums of distances from any point on the ellipse to the two foci are constant.This creates its characteristic oval shape. When the terms \(Ax^2\) and \(Cy^2\) both hold negative signs or one is dominant and they manage to outweigh the middle term \(Bxy\), you achieve this balanced, rounded shape. Moreover, this classification also depends on the relative sizes of \(A\) and \(C\) when \(B = 0\); it ensures that the extension ratios in different axes align to form an ellipse.

Hyperbolas

Hyperbolas appear like two mirrored, open-ended curves. They emerge when a certain condition in an equation involving conic sections is met:- A hyperbola is defined when \(B^2 - 4AC > 0\).- These shapes have a pair of branches, each resembling one side of a parabola, opening in opposite directions.For hyperbolas, the distances are based on a path similar to, yet distinct from, parabolas: Their equation layout often includes competition between \(Ax^2\) or \(Cy^2\) and \(Bxy\). This can reflect a distance or directional shift, pushing the shape into having twin symmetrical paths. Remember, as in our example, a conic section with a positive discriminant like 24 confirms it's a hyperbola.