Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$y^{2}=\frac{14}{3}-x^{2}$$

Step-by-Step Solution

Answer

The graph of the equation is a circle.

Step 1: Recognize the Conic Sections

There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Equations of conic sections can typically be identified based on the degrees and combination of the terms with squares (i.e., either \(x^2\) or \(y^2\)).

Step 2: Analyze the Equation

The given equation is \(y^2 = \frac{14}{3} - x^2\). Notice that there are \(y^2\) and \(x^2\) terms. Specifically, we have a term \(y^2\) on the left and \(-x^2\) on the right.

Step 3: Rearrange the Equation

Moving the \(x^2\) to the other side, the equation becomes \(y^2 + x^2 = \frac{14}{3}\). This looks like a specific form of a circle's equation.

Step 4: Identify the Conic Section Type

A circle in its standard form is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. The equation \(y^2 + x^2 = \frac{14}{3}\) matches this standard form of a circle with the constant \(\frac{14}{3}\) representing \(r^2\).

Step 5: Conclusion

The equation \(y^2 = \frac{14}{3} - x^2\) describes a circle when rearranged into the form \(x^2 + y^2 = \frac{14}{3}\). Therefore, the graph represents a circle.

Key Concepts

Circle Equation

A circle is one of the simplest forms of conic sections in geometry. The standard equation of a circle is found as follows:

• The equation takes the form: \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.
• In this equation, the origin \((0, 0)\) serves as the center of the circle unless otherwise specified.

The given problem asks us to identify the type of conic section represented by the equation \(y^2 = \frac{14}{3} - x^2\). By rearranging the equation to \(y^2 + x^2 = \frac{14}{3}\), we see it now matches the standard circle equation form. This means that the terms \(x^2\) and \(y^2\) describe a circle with a center at the origin, and \(\frac{14}{3}\) as \(r^2\). Thus, the radius \(r\) of the circle is \(\sqrt{\frac{14}{3}}\). Understanding this rearrangement is the key to recognizing the equation's form as that of a circle.

Conic Section Identification

Conic sections are curves obtained by intersecting a cone with a plane. The most common conic sections include circles, ellipses, parabolas, and hyperbolas. To identify which conic section an equation represents, look closely at the squared terms:

• Equations containing both \(x^2\) and \(y^2\) terms generally indicate either a circle, ellipse, or hyperbola.
• When the coefficients of \(x^2\) and \(y^2\) have the same sign and are equal, it's typically a circle.
• If the coefficients are different, it's probably an ellipse.
• Should the coefficients have opposite signs, the equation could be a hyperbola.

In our provided exercise, the equation \(y^2 = \frac{14}{3} - x^2\) contains both \(x^2\) and \(y^2\) terms. Upon rearranging as \(y^2 + x^2 = \frac{14}{3}\), we notice the coefficients of \(x^2\) and \(y^2\) are equal (both are 1), thus identifying this conic section as a circle.

Conic Sections Analysis

Analyzing conic sections involves examining the structure of their equations to better understand their graphical representation. The exercise equation \(y^2 = \frac{14}{3} - x^2\) let's break it down step-by-step:

• The presence of both \(x^2\) and \(y^2\) implies symmetry in both axes, a common feature in circles.
• Moving terms can help us rewrite it into the standard form \(x^2 + y^2 = r^2\), crucial for identifying circles and their geometric properties.

By converting the equation into its standard circle form, one can clearly observe symmetrical properties about the origin, disclosing that:

• The constant term on the right side \(\frac{14}{3}\) is crucial as it equals \(r^2\). It defines the size of the circle.
• A conic section that has equal coefficients for \(x^2\) and \(y^2\) with no additional linear coefficients of \(x\) or \(y\) typically centers at the origin with the equation directly describing it.

Analysis of these terms and inclusion of context via realignment reveals a comprehensive depiction of the geometric shape, facilitating related calculus and algebra tasks.