Problem 30 - Algebra and Trigonometry

Question

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+4 y^{2}+20 x-40 y+300=0 $$

Step-by-Step Solution

Answer

The equation represents a degenerate conic with no graph.

Step 1: Rearrange the Terms

Start with the given equation: \[ x^{2} + 4y^{2} + 20x - 40y + 300 = 0 \]Rearrange the terms by grouping the \(x\) and \(y\) terms together:\[ (x^{2} + 20x) + (4y^{2} - 40y) + 300 = 0 \]

Step 2: Complete the Square for x-terms

For the \(x\) terms, complete the square:Take \(x^{2} + 20x\). To complete the square:1. Take half of the coefficient of \(x\) (which is 20), divide by 2 to get 10, and square it to get 100.2. Add and subtract this square inside the equation:\[ (x^{2} + 20x + 100) - 100 \]

Step 3: Complete the Square for y-terms

For the \(y\) terms, complete the square:Start with \(4y^{2} - 40y\). Factor out 4 first:\[ 4(y^{2} - 10y) \]Then complete the square inside the parentheses:1. Take half of the coefficient of \(y\) (which is -10 inside the parentheses), divide by 2 to get -5, and square it to get 25.2. Add and subtract this square inside the parentheses:\[ 4(y^{2} - 10y + 25 - 25) \]This simplifies to:\[ 4((y - 5)^{2} - 25) \] which is equivalent to\[ 4(y - 5)^{2} - 100 \]

Step 4: Simplify the Equation

Insert the completed squares back into the equation:\[ (x + 10)^{2} - 100 + 4(y - 5)^{2} - 100 + 300 = 0 \]Combine the constants:\[ (x + 10)^{2} + 4(y - 5)^{2} + 100 = 0 \]Subtract 100 from both sides:\[ (x + 10)^{2} + 4(y - 5)^{2} = -100 \]

Step 5: Analyze the Conic

Since the equation \[ (x + 10)^{2} + 4(y - 5)^{2} = -100 \]has a negative value on the right side, this represents a degenerate conic. This equation cannot have a graph in the real plane because the squared term cannot equal a negative number.

Key Concepts

Completing the Square

Completing the square is a mathematical method used to transform a quadratic equation into a perfect square trinomial. This technique is especially helpful when dealing with conic sections, as it can simplify equations to a standard form. For any quadratic term, like \(x^2 + bx\), follow these steps to complete the square:

• Take the coefficient of the linear term, \(b\).
• Divide it by 2 to get \(b/2\), and square the result to \((b/2)^2\).
• Add and subtract \((b/2)^2\) inside the equation to maintain balance.

For example, with \(x^2 + 20x\), you take half of 20, which is 10, and square it to get 100, forming \((x + 10)^2 - 100\). This technique is central to solving problems involving conic sections by making the equation easier to analyze and interpret.

Degenerate Conic

A degenerate conic is a special case in the study of conic sections. It occurs when the conic does not form a typical curve, like an ellipse, parabola, or hyperbola. Instead, the equation does not result in a usual graph when plotted.

• One sign of a degenerate conic is when the equation equals a negative value after completing the square, as squared terms cannot be negative in real numbers.
• Common types of degenerate conics include a single point, a line, or two intersecting lines.

In the solution provided, the equation simplifies to \( (x + 10)^2 + 4(y - 5)^2 = -100 \), indicating a degenerate conic. The negative value on the right side means that the equation has no graph in the real plane.

Ellipse

An ellipse is a type of conic section characterized by its elongated circle shape. It is defined by the equation \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), where \( (h, k) \) is the center.

• The lengths of the major and minor axes are determined by \(2a\) and \(2b\), respectively.
• The foci are located along the major axis at \(c = \sqrt{a^2 - b^2}\) from the center.

Although the original equation needed analysis to see if it represented an ellipse, the resulting negative indicates a degenerate conic. Thus, an ellipse was not applicable here, but understanding its properties is essential for other conic-equation contexts.

Hyperbola

A hyperbola is another conic section with two distinct curves, similar to two mirrored parabolas. It follows a general form of \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) or vice-versa.

• The center is \( (h, k) \), with vertices along the transverse axis at \( \pm a \) from the center.
• The foci, located further from the center, are \( \pm c \), where \(c = \sqrt{a^2 + b^2}\).
• The asymptotes, lines that the hyperbola approaches but never touches, are given by \(y = \pm \frac{b}{a}(x - h) + k\).

In some cases, analyzing the equation might suggest a hyperbola if completed squares result in a comparative form like above. Although the solution initially aimed to check for potential hyperbola, the discovery of a negative value confirmed a degenerate conic instead.