Problem 43 - 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. $$ x=1+2 \sqrt{1-\frac{(y+2)^{2}}{9}} $$

Step-by-Step Solution

Answer

This is the right half of an ellipse centered at (1, -2) with the equation \( \frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1 \).

Step 1: Recognize the format of the standard ellipse equation

The standard form of an ellipsoidal equation is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). This given exercise, however, seems to have a derived part of an ellipse.

Step 2: Rearrange the given equation

Given the equation: \( x = 1 + 2\sqrt{1 - \frac{(y+2)^2}{9}} \). To find the full equation of the ellipse, we will need to manipulate this to fit the standard ellipse format or identify the values of \(a\), \(b\), \(h\), and \(k\).

Step 3: Express the isolated component of the ellipse

Begin by subtracting 1 from both sides: \( x - 1 = 2\sqrt{1 - \frac{(y+2)^2}{9}} \). Then, divide by 2 to isolate the square root: \( \frac{x-1}{2} = \sqrt{1 - \frac{(y+2)^2}{9}} \).

Step 4: Square both sides to eliminate the square root

Square both sides of the equation: \( \left(\frac{x-1}{2}\right)^2 = 1 - \frac{(y+2)^2}{9} \).

Step 5: Rearrange to identify ellipse equation

Rearrange to find the structure of the ellipse: \( \frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1 \). This now fits the standard form of an ellipse centered at \((1, -2)\) with semi-major axis \( b = 3 \) (vertical) and semi-minor axis \( a = 2 \) (horizontal).

Step 6: Determine which half of the ellipse is represented

The relationship \( x = 1 + 2\sqrt{1 - \frac{(y+2)^2}{9}} \) indicates positive square root addition to 1. This means \( x \geq 1 \), so we're talking about the right half of the ellipse.

Key Concepts

Understanding the Standard Ellipse Form

The standard ellipse form is a fundamental equation in geometry that represents the shape and position of an ellipse on the Cartesian plane. It is expressed as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). This equation tells us several things by looking at it:

• \( (h, k) \) is the center of the ellipse.
• \( a \) is the semi-major axis when greater than \( b \), determining the longer stretch of the ellipse.
• \( b \) is the semi-minor axis, pointing to the shorter stretch when \( a > b \).

This form provides a precise method to reframe complex equations into a simpler, recognized template, allowing easier analysis and understanding of the ellipse's properties. When any equation involves squares of \( x \) and \( y \), it is wise to check if it resembles or can be shaped into this standard form.

Semi-Major Axis Demystified

The semi-major axis is a significant characteristic of an ellipse. In simple terms, the semi-major axis is half of the longest diameter across the ellipse. In the standard ellipse equation, represented by \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the semi-major axis is denoted by \( a \) if \( a > b \).For our ellipse equation \( \frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1 \), since \( 9 > 4 \), this makes the \( b = 3 \) the length of our semi-major axis. Thus, the ellipse stretches more prominently in the vertical direction from the center \((1, -2)\). This understanding helps in visualizing the dimensions and layout of the ellipse on a graph.

Mastering Equation Rearrangement

Equation rearrangement involves skillfully manipulating and transforming a complex equation into a recognizable and manageable form. When dealing with an ellipse equation that isn't in standard form, adjusting the equation by isolating terms and using algebraic techniques is key. In the original problem given by \( x = 1 + 2 \sqrt{1-\frac{(y+2)^2}{9}} \), we inconveniently begin outside the standard form. To transform it into \( \frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1 \), we:

• Subtract 1 and divide by 2 to isolate the square root.
• Square both sides to eliminate the square root, removing the radical and simplifying the relationship.
• Adjust the remaining terms to align with the standard form.

Rearranging techniques like these are indispensable skills for effectively tackling mathematical expressions.

Graph Analysis of the Transformed Ellipse

Graph analysis involves examining the characteristics of an equation's visual representation on a coordinate plane. For an ellipse, understanding the graph involves identifying its center, axes, and orientation. Looking at the rearranged ellipse equation, \( \frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1 \), we discover:

• Centered at \( (1, -2) \), obtained by shifting horizontally and vertically.
• Vertically oriented as \( b > a \), meaning the longer dimension aligns along the y-axis.
• Its semi-major axis extends 3 units upwards (and downwards) from the center, while the semi-minor extends 2 units sideways.
• Only the right half represented, due to the specified form of the original equation \( x = 1 + 2\sqrt{...} \).

Analyzing such traits enables accurate plotting and full comprehension of the ellipse's behavior on a graph.