Problem 65 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$\frac{x^{2}}{8}+\frac{y^{2}}{8}=2$$

Step-by-Step Solution

Answer

The graph is a circle with center (0,0) and radius 4.

Step 1: Recognize the Graph Type

The given equation \( \frac{x^2}{8} + \frac{y^2}{8} = 2 \) is in the form of an ellipse. The form for an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Step 2: Rewrite the Equation

To put the equation in the standard form of an ellipse, divide the whole equation by 2:\[\frac{x^2}{16} + \frac{y^2}{16} = 1\] Now the equation is in the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \( a^2 = 16 \) and \( b^2 = 16 \).

Step 3: Identify the Center, Axes and Radius

Since it's an ellipse and both denominators are equal (\( a^2 = b^2 = 16 \)), it is actually a circle. The center of the circle is at the origin, \((0, 0)\), and the radius is \( \sqrt{16} = 4 \).

Step 4: Sketch the Graph

To sketch the graph, draw a circle centered at the origin (0, 0) with a radius of 4. Each point on this circle is located 4 units from the center in all directions.

Key Concepts

Conic Sections

Conic sections are a fascinating topic in geometry, offering a glimpse into how slicing a cone at different angles can produce various shapes. These shapes include circles, ellipses, parabolas, and hyperbolas.
When you intersect a plane with a cone parallel to its base, we get a circle. This special case of an ellipse gives us the simplest conic section, where the two axes are equal.
If the plane is at a slight angle to the base, the section results in an ellipse, where typically the lengths of the major and minor axes differ. Adjusting the angle further can produce a parabola or hyperbola, each with its own unique properties and equations.

• **Circle:** The set of all points in a plane that are equidistant from a given point, called the center.
• **Ellipse:** Extends the idea of a circle, with two focal points.
• **Parabola:** Formed when a plane is parallel to the edge of the cone.
• **Hyperbola:** Occurs when a plane intersects both halves of a double cone.

Understanding these structures is crucial, as they form the basis of many real-world phenomena and demonstrate key principles in mathematics.

Circle Graph

When talking about a circle graph in terms of conic sections, we're referring to a representation of the circle equation on a coordinate plane. For circles, the equation can generally be expressed as \( (x-h)^2 + (y-k)^2 = r^2 \). Here,

• \( (h, k) \) represents the center of the circle.
• \( r \) represents the radius of the circle.

In the problem we're examining, after simplifying the ellipse equation, \( \frac{x^2}{16} + \frac{y^2}{16} = 1 \), it is identified as a circle due to both \( a^2 \) and \( b^2 \) being equal. The circle graph portrays these equal axes, indicating a symmetrical shape centered around the origin, \((0, 0)\).
To graph it, you would place the center at (\( 0, 0 \)) and draw out a circle with its radius extending in all directions by the given value. In this case, that value is 4, identified from the square root of 16. Circle graphs are not just a fundamental component of geometry, but also helpful in visualizing data in a compelling way.

Center and Radius

The concepts of the center and radius are key components when describing and working with circles, particularly in equations. In mathematics, they define the essential features of a circle and help in sketching it accurately on a graph.
The center of a circle is the fixed point from which all points on the circle are evenly spaced. In coordinate geometry, it is commonly denoted as \( (h, k) \). This tells us the exact location of the circle on a Cartesian plane.
The radius, on the other hand, is the constant distance from the center to any point on the circle's edge. It's denoted by \( r \), and in equations, it appears in squared form, \( r^2 \), usually equated to the sum of squared distances from the center to a point on the edge.
In the solution provided for the initial equation \( \frac{x^2}{8} + \frac{y^2}{8} = 2 \), by transforming it to the form \( \frac{x^2}{16} + \frac{y^2}{16} = 1 \), it is shown that the center is \((0, 0)\) and the radius is \(4\). This standard format allows anyone to easily determine these critical parameters, making it simpler to understand and visualize the circle.