Problem 28 - Algebra and Trigonometry

Question

\(23-34\) 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 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. $$ 2 x^{2}+y^{2}=2 y+1 $$

Step-by-Step Solution

Answer

The equation represents an ellipse with center \((0,1)\), major axis \(\sqrt{2}\), minor axis 1.

Step 1: Rearrange the Equation

Start by moving all terms involving variables to one side of the equation. Rewrite the original equation: \[ 2x^2 + y^2 - 2y = 1 \]

Step 2: Complete the Square for the y-Term

Focus on the terms involving \(y\). Take \(y^2 - 2y\), and complete the square:\(y^2 - 2y = (y-1)^2 - 1\).Substitute it back into the equation:\[ 2x^2 + (y-1)^2 - 1 = 1 \].

Step 3: Simplify the Equation

Add 1 to both sides to simplify the equation:\[ 2x^2 + (y-1)^2 = 2 \].

Step 4: Identify the Conic Section

Divide every term by 2 to compare with standard form:\[ \frac{x^2}{1} + \frac{(y-1)^2}{2} = 1 \].This equation is of the form \(\frac{x^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), indicating an ellipse because both coefficients are positive and different.

Step 5: Determine the Center and Axes

The center of the ellipse is at \((0, 1)\). The lengths of the axes are determined by the denominators in the equation. Here, the major axis has length \( \sqrt{2} \) (in the y-direction), and the minor axis has length \(1\) (in the x-direction).

Step 6: Find the Foci

The distance to the foci \(c\) is calculated using the formula \(c = \sqrt{b^2 - a^2}\): \[ c = \sqrt{2 - 1} = 1 \]. Thus, the foci are located at \((0, 1+1)\) and \((0, 1-1)\) which are \((0, 2)\) and \((0, 0)\).

Step 7: Sketch the Graph

Draw the ellipse centered at \((0, 1)\). The minor axis is aligned along the x-axis with length \(2\), and the major axis is along the y-axis with length \(2\sqrt{2}\). Mark the vertices and foci on the graph.

Key Concepts

Ellipse Equations

An equation representing an ellipse takes the standard form\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]where \((h, k)\) is the center of the ellipse. The terms \(a\) and \(b\) represent the distances from the center to the ellipse along its axes. The larger of \(a\) or \(b\) determines the "major" axis, while the smaller determines the "minor" axis. Notably, if these distances are equal, the ellipse is a circle.
This equation requires completing the square to identify \((h, k)\) and figure out which axis is major or minor. An ellipse always shows up when each squared term is positive and coefficients differ, indicating a stretched circular shape along one axis.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. They consist of four types: ellipses, parabolas, hyperbolas, and circles (a special case of an ellipse).
A key feature distinguishing them is their standard forms:

• Ellipses bend symmetrically around two axes, remaining bounded in both directions.
• Parabolas open outward in a U-shape along one direction.
• Hyperbolas have two branches opening in opposite directions.

An equation in the elliptical form will have both squared terms positive and coefficients unequal. This unique characteristic helps differentiate ellipses from other conics.

Foci of Ellipse

The foci of an ellipse are two distinct points lying along its major axis, symmetrically apart from its center. These points hold the unique property that any point on the ellipse is the same total distance from both foci.
To locate them, use the formula for distance \(c\), given as \(c = \sqrt{|b^2 - a^2|}\). Depending on which axis is longer, this modifies either the x or y values by \(c\).
Focusing on the foci helps understand an ellipse's stretch because they dictate the overall shape. The closer the foci to the center, the more circular the ellipse appears.

Major and Minor Axes

The axes of an ellipse are vital for understanding its orientation and dimensions. They refer to two perpendicular lines passing through the center, representing the maximum and minimum extents of the shape.

• The "major axis" is the longest diameter, spanning from vertex to vertex, representing the wider side.
• The "minor axis" is shorter, noting the thinner side.

Length of the major axis is calculated as \(2a\), and the minor as \(2b\). Identifying these axes is crucial in graphing and explaining how the ellipse is stretched or compressed.