Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=-2-5 \sqrt{1-\frac{(y-1)^{2}}{16}} $$

Step-by-Step Solution

Answer

The graph represents the left half of an ellipse: \( \frac{(x+2)^2}{25} + \frac{(y-1)^2}{16} = 1 \).

Step 1: Understanding the Equation

The given equation is \(x = -2 - 5 \sqrt{1 - \frac{(y-1)^2}{16}}\). This equation resembles the form of an ellipse in its derivation from the standard equation \(y = h \pm b \sqrt{1 - \frac{(x-k)^2}{a^2}}\), but here is rearranged in terms of \(y\). The term under the square root indicates the condition of an ellipse since \( \frac{(y-1)^2}{16} \) is inside the square root.

Step 2: Reconstructing the Ellipse Equation

Express the provided equation in standard ellipse form by solving for the identical components. Start by recognizing the terms: 5 can be related to \(a\) as in the ellipse formula \(x = k \pm a \sqrt{1 - \frac{(y-h)^2}{b^2}}\), thus equating \(5\) to \(a\), and 16 corresponds to \(b^2\). Rewriting, the standard form becomes \( \frac{(x+2)^2}{25} + \frac{(y-1)^2}{16} = 1 \), which rearranges into a typical ellipse equation.

Step 3: Identifying the Half of Ellipse

Since the given equation includes \(x = -2 - 5 \sqrt{1 - \frac{(y-1)^2}{16}}\), it represents the left side due to the negative square root. The subtraction signifies that \(x\) decreases as \(y\) values satisfy \((y-1)^2 < 16\). Hence, the graph represents the left half of an ellipse.

Key Concepts

Ellipse Properties

An ellipse is a very interesting shape often regarded as an elongated circle. Understanding its properties is key to exploring conic sections. The standard form of an ellipse's equation in Cartesian coordinates is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center of the ellipse. The terms \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively.

• If \(a > b\), the ellipse is elongated along the x-axis, making it wider horizontally.
• If \(b > a\), it is stretched vertically along the y-axis.
• The endpoints of the major and minor axes are called vertices.
• Each \'focus\' of an ellipse lies on the major axis and helps define the shape geometrically.

Knowing these properties can help immensely when you are tasked with writing or interpreting the equation of an ellipse. The presence of a square root term and a fractional \'denominator\' in the expression clues you in on the existence of elliptical properties in equations.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four primary types of conic sections are circles, ellipses, parabolas, and hyperbolas, each with distinctive forms and characteristics.
Ellipses emerge when the intersecting plane cuts through the cone at an angle less than the slope of the cone but does not pass through the apex. This results in a closed curve, distinct from other conics that can be open. In mathematical contexts, conic sections reflect various real-world applications ranging from orbit paths to architecture.

• Circle: A special case of an ellipse where \(a = b\).
• Parabola: Forms when the plane is parallel to the slant height of the cone.
• Hyperbola: Created when the plane cuts through both halves of the cone.

Understanding conic sections provides deeper insights into the behavior and equations of each shape, specifically how they rotate and intersect in space.

Graphing Ellipses

Graphing an ellipse accurately requires understanding its equation and features. Start by determining the center, which for the general form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), is given by \((h, k)\). Next, identify the lengths of the axis.

• Major Axis: Defined as \(2a\) units long, stretching across the ellipse's longest part.
• Minor Axis: Spanning \(2b\) units, this line passes through the shorter dimension.

To graph, mark the center, then measure \(a\) units along the major axis and \(b\) units along the minor. Plot these points to outline the ellipse. Focus also on the directionality defined by the axes.
In exercises where only a half of the ellipse is drawn, as in the given problem, understanding direction is crucial for the equation manipulation. For example, a negative in part of the equation often indicates a targeted half, such as the left or lower portion of a graph.