Problem 47 - Precalculus: Functions and Graphs

Question

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. $$\text { Left half of } \frac{x^{2}}{36}+\frac{y^{2}}{25}=1$$

Step-by-Step Solution

Answer

Equation: \(\frac{x^2}{36} + \frac{y^2}{25} = 1\), with \(x \leq 0\).

Step 1: Understand the Standard Form of an Ellipse

The given equation \(\frac{x^2}{36} + \frac{y^2}{25} = 1\) is in the standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 36\) and \(b^2 = 25\). This tells us the ellipse is centered at the origin \((0,0)\) and has a semi-major axis of length 6 along the x-axis and a semi-minor axis of length 5 along the y-axis.

Step 2: Identify the Half of the Ellipse Required

We are asked to find the equation for the left half of the ellipse. The left half refers to the section of the ellipse where the x-values are negative, i.e., \(x \leq 0\).

Step 3: Modify the Equation for the Left Half

To define only the left half, we keep the ellipse's equation as it is algebraically but restrict \(x\) to \(x \leq 0\). So, the equation for the left half of the ellipse is still \(\frac{x^2}{36} + \frac{y^2}{25} = 1\) with the condition \(x \leq 0\).

Key Concepts

half of an ellipse

When dealing with ellipses, it's important to understand that an ellipse can be split into two halves. Either horizontally or vertically, based on its orientation. Looking at the given problem, we are interested in the 'left half' of the ellipse. Conceptually, this means focusing only on the part of the ellipse that exists on one side of the axis. In this exercise, the left half indicates that we will be considering the portion of the ellipse where all the x values are less than or equal to zero.

To find this part, we do not need to change the original equation. Instead, the focus is on restricting the x values by specifying a condition: **x ≤ 0**. This ensures that we only look at half of the shape.

• The equation remains the same: \(\frac{x^2}{36} + \frac{y^2}{25} = 1\).
• However, remember: restrict to \(x \leq 0\).

This approach works because the condition naturally eliminates the right half, effectively giving us the left half of the ellipse without altering the algebraic expression.

ellipse centered at origin

An ellipse centered at the origin means that its center point is at the coordinate \(0,0\). This is an important aspect, as it simplifies our calculations and understanding of the shape's orientation and distribution. For the given exercise, the ellipse's equation \(\frac{x^2}{36} + \frac{y^2}{25} = 1\) confirms that the center is at the origin.

Ellipses centered at the origin are symmetrical about both the x-axis and y-axis. This fact makes it easier to identify and define halves of the ellipse. For example, in this scenario, identifying the left half simply involves considering the x values on the left side (negative side) of the y-axis. There is no need to visually adjust or move the shape as the symmetry around the origin helps with manipulation and understanding.

• Center of ellipse: \(0,0\)
• Symmetrical across both axes.
• Easy to describe sections, like the left half.

Whenever you encounter an ellipse with an origin-centered orientation, remember how this symmetry benefits simplifying your work.

standard form of an ellipse

The standard form of an ellipse is a foundational concept in understanding and graphing these shapes. The ellipse's equation is given as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, **a** and **b** represent the lengths of the semi-major and semi-minor axes respectively. For the equation \(\frac{x^2}{36} + \frac{y^2}{25} = 1\), this tells us several important things:

• \(a^2 = 36\), so \(a = 6\)
• \(b^2 = 25\), so \(b = 5\)

This configuration defines the size and stretching of the ellipse along the axes. For this specific ellipse, **a** is greater than **b**, indicating that the major axis is longer along the x-direction. This provides us with a horizontal ellipse.

Understanding the standard form allows you to quickly derive key characteristics of any ellipse, such as orientation and axis lengths, which are vital in tasks like identifying halves or sections of the ellipse. The symmetry and simplicity of ellipses in their standard form make them manageable and predictable, especially when combined with an origin-centered positioning.