Problem 49 - Precalculus: Functions and Graphs

Question

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. Upper half of \(x^{2}+3 y^{2}=17\)

Step-by-Step Solution

Answer

The equation for the upper half of the ellipse is \( y = \sqrt{\frac{17 - x^2}{3}} \).

Step 1: Understand the given equation

The given equation is of an ellipse, which is characterized by the general form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In this case, the equation is \( x^{2} + 3y^{2} = 17 \). This indicates an ellipse centered at the origin (0,0).

Step 2: Rewrite the equation in standard form

To transform the equation into standard form, divide all terms by 17: \( \frac{x^2}{17} + \frac{y^2}{\frac{17}{3}} = 1 \). This corresponds to an ellipse with semi-major axis \( a^2 = 17 \) and semi-minor axis \( b^2 = \frac{17}{3} \).

Step 3: Determine the upper half condition

The upper half of the ellipse is represented by only considering the positive values of \(y\). This simplifies the equation since \(y\) is expressed in terms of \(x\).

Step 4: Solve for y in terms of x

From \( x^{2} + 3y^{2} = 17 \), isolate \( y^2 \) to get \( 3y^{2} = 17 - x^{2} \), then \( y^{2} = \frac{17 - x^{2}}{3} \). Consequently, \( y = \sqrt{\frac{17 - x^{2}}{3}} \). This represents the upper half of the ellipse.

Key Concepts

Standard Form of Ellipse

An ellipse is a type of conic section that looks like a stretched circle. The equation for an ellipse is not always provided in its standard form. The standard form helps us easily identify the ellipse's characteristics, such as its axes. The standard form of an ellipse centered at the origin is given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

In this equation, \(a\) represents the semi-major axis, and \(b\) is the semi-minor axis. To convert an equation like \(x^2 + 3y^2 = 17\) into this standard form, you divide each term by 17:

• \( \frac{x^2}{17} + \frac{y^2}{17/3} = 1 \)

This division ensures that the right side of the equation equals 1, a requirement for the standard form.

Semi-Major Axis

The semi-major axis of an ellipse is the longest radius that extends from the center to the edge of the ellipse. In the standard form of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the term \(a^2\) corresponds to the length of the semi-major axis squared.
For the example equation \( \frac{x^2}{17} + \frac{y^2}{17/3} = 1 \), we identify that \(a^2 = 17\). The length of the semi-major axis, \(a\), is then obtained by taking the square root of \(a^2\):

• \( a = \sqrt{17} \)

This axis is longer than the semi-minor axis, as reflects an important feature of the ellipse.

Semi-Minor Axis

The semi-minor axis is the shorter radius stretching from the center of the ellipse to the edge. It's perpendicular to the semi-major axis. In the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \(b^2\) represents the square of the semi-minor axis length.
For the equation transformed into the standard form, \( b^2 \) is given as \( \frac{17}{3} \). To find the length, \(b\), take the square root of \( b^2 \):

• \( b = \sqrt{\frac{17}{3}} \)

This smaller dimension helps us understand the "narrowness" of the ellipse, which results from the shorter radius.

Equation Manipulation

When given an equation that does not initially appear in standard form, some manipulation is necessary to understand its properties. Manipulating the equation involves rearranging terms or dividing, to fit the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
For our example, we start with \(x^2 + 3y^2 = 17\). To convert this to standard form, we divide every term by 17:

• \( \frac{x^2}{17} + \frac{3y^2}{17} = 1 \)

Further, simplify the coefficient for \(y^2\) by factoring the 3:

• \( \frac{x^2}{17} + \frac{y^2}{17/3} = 1 \)

Such manipulation makes it easier to deduce the ellipse's key dimensions and understand its graph.