Problem 54 - Algebra 2

Question

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ -3 x^{2}+x y+2 y^{2}+4 x-7 y=0 $$

Step-by-Step Solution

Answer

A = -3, B = 1, C = 2.

Step 1: Identify the coefficients of x², xy, and y²

In the given equation \[-3x^2 + xy + 2y^2 + 4x - 7y = 0, \]identify the terms that involve the products of the variables x and y. These are:- The coefficient of \(x^2\) is \(-3\).- The coefficient of \(xy\) is \(1\) (since it's \(xy\)).- The coefficient of \(y^2\) is \(2\).

Step 2: Assign the coefficients to A, B, and C

The general form equation is\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.\]From Step 1, we assign:- \(A = -3\)- \(B = 1\)- \(C = 2\)

Key Concepts

Quadratic equations

Quadratic equations are fundamental in algebra and arise frequently when studying the geometry of conic sections, such as circles, ellipses, parabolas, and hyperbolas. A quadratic equation is characterized by its degree, which is 2, meaning that the highest power of the variable is squared. This makes the basic form of a quadratic equation appear as \(ax^2 + bx + c = 0\). It is important to note that all terms should be present in the equation, and if any term is missing, its coefficient is understood to be zero.
In the context of conic sections, we often encounter quadratic equations in two variables, such as \(-3x^2 + xy + 2y^2 + 4x - 7y = 0\). Such equations define shapes in the coordinate plane. When solving quadratic equations, the objective is to find the variable values that satisfy these equations, otherwise termed as the roots or solutions. The key challenge lies in identifying these roots, particularly when the quadratic equation describes complex shapes.

Coefficients

Coefficients are the numerical factors that multiply the variables in an equation, particularly a quadratic equation. These coefficients play a critical role in determining the shape and orientation of a conic section. Let's break down their significance in our example equation:\(-3x^2 + xy + 2y^2 + 4x - 7y = 0\).

• **The coefficient of \(x^2\)**, \(A = -3\): This denotes the concavity of the shape with respect to the x-axis. A negative coefficient suggests that the curve opens downwards if standalone.
• **The coefficient of \(xy\)**, \(B = 1\): This term introduces a shear or rotation in the conic section, altering its orientation and not just its shape.
• **The coefficient of \(y^2\)**, \(C = 2\): Similar to \(A\), this dictates how the curve behaves along the y-axis. A positive value typically suggests the curve opens upwards if isolated.

Understanding these coefficients aids in visualizing the graph of the equation and determining its classification, whether it's a hyperbola, ellipse, or another conic section.

General quadratic form

The general quadratic form is a standard approach to writing equations involving two variables that govern the shape and properties of conic sections. It is represented as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Each component in this formula holds specific information about the geometric construction it describes.
The essence of the general quadratic form is in how it encapsulates various geometric transformations. Here's what the main components imply:

• **Terms \(Ax^2\) and \(Cy^2\)** represent the "pure" quadratic terms affecting width and height, respectively.
• **The \(Bxy\) term** often affects rotations and shearing. Its presence indicates that the ellipse or hyperbola is tilted away from the standard axes.
• **Linear terms \(Dx\) and \(Ey\)** are responsible for shifting the conic section away from the origin along the x and y axes.
• **The constant term \(F\)** adjusts the overall size and position but doesn't affect the shape of the conic.

Mastery of the general quadratic form enables a clear understanding of complex symmetrical shapes and their transformations in coordinate geometry. This is a vital foundation for exploring more advanced topics in mathematics.