Problem 39 - Algebra and Trigonometry

Question

Determine what the value of \(F\) must be if the graph of the equation $$4 x^{2}+y^{2}+4(x-2 y)+F=0$$ is (a) an ellipse, (b) a single point, or (c) the empty set.

Step-by-Step Solution

Answer

(a) For an ellipse: \(F < 17\). (b) For a single point: \(F = 17\). (c) For the empty set: \(F > 17\).

Step 1: Expand and Rearrange the Equation

First, take the original equation \(4x^2+y^2+4(x-2y)+F=0\) and expand it to \(4x^2 + 4x + y^2 - 8y + F = 0\). This can be rearranged into \(4x^2 + 4x + y^2 - 8y = -F\).

Step 2: Complete the Square for x-terms

Complete the square for the \(x\)-terms: \(4x^2 + 4x\). Factor out the 4: \(4(x^2 + x)\). The expression inside the parentheses can be completed to \((x + \frac{1}{2})^2 - \frac{1}{4}\). Hence, \(4(x^2 + x)\) becomes \(4((x+\frac{1}{2})^2-\frac{1}{4}) = 4(x+\frac{1}{2})^2 - 1\).

Step 3: Complete the Square for y-terms

Complete the square for the \(y\)-terms: \(y^2 - 8y\). This can be expressed as \((y-4)^2 - 16\).

Step 4: Substitute Completed Squares

Substitute the completed squares back into the equation, yielding: \[4(x+\frac{1}{2})^2 - 1 + (y-4)^2 - 16 + F = 0\]. Simplify it to \[4(x+\frac{1}{2})^2 + (y-4)^2 = 17 - F\].

Step 5: Determine Values of F for Each Condition

For an ellipse, the right side must be positive, i.e., \(17 - F > 0\), so \(F < 17\). For a single point, the right side of the equation must be zero, i.e., \(17 - F = 0\), so \(F = 17\). For the empty set, the right side must be negative, i.e., \(17 - F < 0\), so \(F > 17\).

Key Concepts

Ellipse

An ellipse is a closed curve that looks somewhat like a stretched circle. It is formed by a set of points in a plane. When added together, the distances from any point on the ellipse to two fixed points (the foci) is constant. Ellipses are an important part of conic sections — shapes that can be created by intersecting a cone with a plane. Ellipses have many practical applications such as orbits of planets and the design of lenses.In algebra, an ellipse can be represented in standard form by its equation:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where

• \(h, k\) are the coordinates of the center of the ellipse.
• \(a\) and \(b\) are the distances from the center to the vertices along the x and y axes respectively.

An ellipse is specified by this positive sum indicating the total distances involved in its shape. In understanding the problem, when determining if the equation represents an ellipse, it's necessary to ensure the expression is positive, specifically \(17 - F > 0\), or \(F < 17\). Clarifying the relationship between these elements allows students to better understand how changes in equations affect the shape of the ellipse.

Completing the Square

Completing the square is a vital technique used in algebra to transform a quadratic expression into a form that is easier to work with. This process involves rewriting a quadratic equation For instance:Consider the quadratic expression \(x^2 + bx\). To complete the square, we follow these steps:

• Add and subtract \((\frac{b}{2})^2\) inside the equation.
• The expression becomes \( (x + \frac{b}{2})^2 - (\frac{b}{2})^2 \).

Now, it is in a perfect square form and easier to manipulate.In terms of the original problem, completing the square helps in rearranging terms that facilitate in identifying different conic sections. Once quadratic terms in both \(x\) and \(y\) are transformed into perfect squares, it's simpler to see the standard forms of conic sections, like ellipses. Step by step, it breaks down complicated algebraic expressions into more manageable parts.

Graphing Equations

Graphing equations allows us to visually interpret mathematical relationships, creating a bridge between algebraic expressions and their graphical counterparts.When graphing an equation, it's important to recognize its form. Each form holds clues as to the type of graph it will produce — whether it is a line, a parabola, or a conic section like an ellipse.

• Begin by writing the equation in a recognizable form, such as the standard form or vertex form.
• Identify the key components like center, radius, stretches, and direction.

In our example, post analysis leads us to the transformed equation:\[ 4(x+\frac{1}{2})^2 + (y-4)^2 = 17 - F \]This equation can represent different conic sections depending on the value of \(F\). When graphing, remember:

• If the right side is positive, the result is an ellipse.
• If zero, it reduces to a point, and if negative, an empty set, making it non-existent in graph form.

Recognizing these forms helps in understanding and predicting the changes in graphical representation.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is the language of mathematics, allowing us to create formulas and solve mathematical problems through operations and procedures.Understanding algebra is crucial for solving complex equations and understanding the properties of numbers and their relationships. It involves:

• Identifying and organizing expressions effectively.
• Applying operations like addition, multiplication, and exponentiation to variables and constants.

In this exercise, algebra is employed to simplify and manipulate the original equation \[ 4x^2 + y^2 + 4(x-2y) + F = 0 \] to explore and determine what values of \(F\) turn it into an ellipse, a point, or an empty set. Completing the square and rewriting expressions showcases the power of algebra in making sense of complex mathematical constructs. This enables students to not just solve problems but also gain deeper insights into mathematical relationships and their applications.