Problem 35 - 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) \(F > 17\), (b) \(F = 17\), (c) \(F < 17\).

Step 1: Start with the General Conic Section Equation

The given equation is a quadratic expression in terms of \(x\) and \(y\): \[ 4x^2 + y^2 + 4x - 8y + F = 0 \] We need to rewrite it to match the standard form of conics.

Step 2: Complete the Square for Terms in x

Focus on \(x\)-related terms: \[ 4x^2 + 4x \] Factor out 4: \[ 4(x^2 + x) \] Complete the square inside the parenthesis:\(x^2 + x = (x + \frac{1}{2})^2 - \frac{1}{4} \)Thus:\[ 4((x + \frac{1}{2})^2 - \frac{1}{4}) = 4(x + \frac{1}{2})^2 - 1 \] This becomes:\( 4(x + \frac{1}{2})^2 - 1 \).

Step 3: Complete the Square for Terms in y

Focus on \(y\)-related terms:\[ y^2 - 8y \]Complete the square:\(y^2 - 8y = (y - 4)^2 - 16\)Thus:\[ (y - 4)^2 - 16 \].

Step 4: Rewrite the Equation

Substitute the completed square forms back into the original equation:\[ 4(x + \frac{1}{2})^2 - 1 + (y - 4)^2 - 16 + F = 0 \]Simplify:\[ 4(x + \frac{1}{2})^2 + (y - 4)^2 + (F - 17) = 0 \].

Step 5: Case (a): Determine F for an Ellipse

For an ellipse, the left-hand side of the equation should equal a positive number:\[ F - 17 > 0 \]Thus, \(F > 17\).

Step 6: Case (b): Determine F for a Single Point

For a single point, both squares must sum to zero, making:\[ F - 17 = 0 \]Thus, \(F = 17\).

Step 7: Case (c): Determine F for an Empty Set

For an empty set, the left-hand side of the equation should be negative:\[ F - 17 < 0 \]Thus, \(F < 17\).

Key Concepts

Ellipse

An ellipse is a type of conic section that resembles an elongated circle or an oval. It is defined as the set of all points where the sum of the distances to two specific points, called foci, is constant.
This means that for any point on the ellipse, the distance to one focus plus the distance to the other focus stays the same. This property gives ellipses their characteristic shape.In a standard ellipse equation, the general form is:

• a\(x^2\) + b\(y^2\) + c\(x\) + d\(y\) + \(F = 0\)

To identify the conic section as an ellipse, inequalities in the standard form must be met, particularly in terms of coefficients of \(x^2\) and \(y^2\). To ensure it forms an ellipse, the condition \(a eq b\) (for different coefficients) is crucial.
In summary, an ellipse's geometric properties, as expressed in its equation, ensure a balanced structure differing from other conics like parabolas and hyperbolas.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This method simplifies the equation, making it easier to analyze or graph.Here's how you complete the square:

• Take a quadratic expression such as \(x^2 + bx\).
• Find half of the coefficient of \(x\), which is \(\frac{b}{2}\), and square it.
• Add and subtract this square inside the expression to transform it into a perfect square trinomial \((x + \frac{b}{2})^2 - \frac{b^2}{4}\).

This process is essential when you need to rewrite a quadratic equation in vertex form, which has a clear vertex, making it simpler to identify the maximum or minimum points.
The method applies not only to quadratics in one variable but also extends to conic sections, like circles and ellipses, helping to standardize their equations towards easier interpretation.

Quadratic Equations

Quadratic equations are polynomials of degree 2, typically written in the form \(ax^2 + bx + c = 0\). These equations have significant applications across mathematics and sciences owing to their predictable solutions and geometric interpretations.Quadratic equations can have:

• Two distinct real roots,
• One real root (when the equation forms a perfect square), or
• No real roots but two complex roots.

Solutions to quadratic equations can be found using the quadratic formula: \(\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]\)
The expression under the square root, \(b^2 - 4ac\), is the discriminant, which determines the nature of the roots.
If this discriminant is positive, there are two distinct real solutions; if it's zero, there is one real solution; and if it's negative, there are two complex solutions.Quadratic equations serve as a foundational element in understanding more complex algebraic structures by acting as a bridge between linear equations and higher-degree polynomials.