Problem 58 - Precalculus: Mathematics for Calculus

Question

Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an cllipse, 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 results in a degenerate conic with no graph.

Step 1: Move the Constant Term

The given equation is \(x^2 + 4y^2 + 20x - 40y + 300 = 0\). First, move the constant term to the other side of the equation by subtracting 300 from both sides.\[ x^2 + 4y^2 + 20x - 40y = -300 \]

Step 2: Group and Rearrange Terms

Group and rearrange the terms to prepare for completing the square:\[ (x^2 + 20x) + (4y^2 - 40y) = -300 \]

Step 3: Complete the Square for x

Complete the square for the \(x\) terms. Take half of the coefficient of \(x\), which is 20, square it, and add/subtract it inside the equation:\[ (x^2 + 20x) = (x + 10)^2 - 100 \]

Step 4: Complete the Square for y

Factor 4 out of the \(y\) terms and complete the square:\[ 4(y^2 - 10y) = 4((y - 5)^2 - 25) = 4(y - 5)^2 - 100 \]

Step 5: Substitute Back and Simplify

Substitute the completed squares back into the equation:\[ (x + 10)^2 - 100 + 4(y - 5)^2 - 100 = -300 \] Then simplify:\[ (x + 10)^2 + 4(y - 5)^2 = -300 + 200 \] which simplifies to:\[ (x + 10)^2 + 4(y - 5)^2 = -100 \]

Step 6: Analyze and Conclude

Since the equation \((x + 10)^2 + 4(y - 5)^2 = -100\) results in a negative right-hand side, the equation does not describe a graphable conic section in the real number plane. Thus, it is a degenerate conic and has no graph.

Key Concepts

Completing the Square

Completing the square is a helpful technique used in algebra to transform a quadratic equation into a perfect square trinomial. This method is essential when dealing with conic sections, as it allows us to rewrite equations in a standardized form. Here's a quick overview of how to complete the square.

• First, identify the quadratic term and the linear term of the variable you want to complete the square for (e.g., for the expression \(x^2 + 20x\), these terms are \(x^2\) and \(20x\)).
• Take half of the coefficient of the linear term and square it (in this case, \(\frac{20}{2} = 10\) and \(10^2 = 100\)).
• Add and subtract this square within the equation to maintain its balance.

This technique is applied separately to both the \(x\) and \(y\) components when dealing with conic sections. This process can reveal much about the conic's properties, such as its shape and position.

Degenerate Conic

A degenerate conic section occurs when a conic section doesn't form a typical shape like a circle, ellipse, parabola, or hyperbola. Instead, it reduces to simpler forms such as a point, a line, or intersecting lines. This might happen when the right-hand side of a transformed equation results in a negative number, leading to no solution in the real number plane.

In the example given, \[(x + 10)^2 + 4(y - 5)^2 = -100\],completing the square resulted in a negative number, which means that the equation cannot satisfy any real values of \(x\) and \(y\). Therefore, this equation represents a degenerate conic, as it does not describe a recognizable conic graph. Understanding the conditions for a degenerate conic is vital when determining the nature of conics within processes such as completing the square.

Ellipse

An ellipse is a type of conic section that resembles a stretched circle. It is formed by intersecting a plane with a cone at an angle such that it creates a closed curve. Ellipses have several distinct features:

• Center: The midpoint of the ellipse.
• Foci: Two points located inside the ellipse that are used in its formal definition.
• Vertices: The most extended points along the major axis of the ellipse.
• Major and minor axes: The longest and shortest diameters of the ellipse, respectively.

The general form equation for an ellipse is \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\],where \((h, k)\) is the center, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes.

Identifying an ellipse from a general quadratic equation requires completing the square to transform the expression into this form. However, in the exercise problem, completing the square does not yield a valid ellipse because of the resulting degenerate conic.

Hyperbola

A hyperbola is another type of conic section formed by intersecting a plane with both halves of a double cone. It consists of two separate, mirror-image curves and has the following characteristics:

• Center: The midpoint between the vertices of the hyperbola.
• Foci: Two points outside the hyperbola used to define it.
• Vertices: Points at the ends of the hyperbola's transverse axis.
• Asymptotes: Lines that the hyperbola approaches but never meets.

The standard form of a hyperbola's equation is \[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\],where \((h, k)\) is the center.

In the given problem, the equation could not be rewritten into the standard form of a hyperbola because it resulted in a degenerate conic with no graph. This distinction is crucial in understanding when an equation cannot represent any real conic section, like a hyperbola, in the geometry of real numbers.