Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Match each equation with the appropriate description . Do not use a calculator. $$x^{2}+y^{2}=5$$ A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward

Step-by-Step Solution

Answer

Option F: Circle; center \((0,0)\); radius \(\sqrt{5}\).

Step 1: Recognize the Equation Type

The given equation is \(x^2 + y^2 = 5\). Recognizing the standard forms of different conic sections can help identify the type of equation. \(x^2 + y^2 = r^2\) is the formula for a circle centered at the origin \((0,0)\).

Step 2: Compare the Equation to the Circle Standard Form

The specific form for a circle with center at \((h,k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). Since \(x^2 + y^2 = 5\) matches \((x-0)^2 + (y-0)^2 = \sqrt{5}^2\), it represents a circle centered at the origin.

Step 3: Identify the Circle's Characteristics

For \(x^2 + y^2 = 5\), the center is \((0,0)\) and the radius is \(\sqrt{5}\), because \(\sqrt{5}\) is the value such that when squared, it yields 5.

Step 4: Match with the Correct Description

Among the given options, F ("Circle; center \((0,0)\); radius \(\sqrt{5}\)") matches the description of the circle derived from the equation.

Key Concepts

Circle Equation

A circle's equation in a coordinate system is always represented by a certain standard form. This form allows us to easily spot the equation of a circle and identify its basic elements, such as its center and radius. In general, the equation for a circle with its center at the origin is written as:\[x^2 + y^2 = r^2\]where \(r\) represents the radius of the circle.

• The equation \(x^2 + y^2 = 5\) is an example of a circle centered at the origin \((0,0)\).
• The expression \(x^2 + y^2\) shows that both \(x\)- and \(y\)-terms are squared, which is a key indicator of a circle in conic sections.

Understanding this concept is crucial to solving more complex problems involving conic sections.

Standard Form

The standard form of a circle equation involves having the squared terms for \(x\) and \(y\) standing alone without any coefficients in a simple form. For a circle with center \((h, k)\) and radius \(r\), the generalized circle equation is:\[(x-h)^2 + (y-k)^2 = r^2\]

• If \(h\) and \(k\) are zero, the circle is centered at the origin, so the equation simplifies to \(x^2 + y^2 = r^2\).
• A comparison between equations, like \(x^2 + y^2 = 5\) and the standard form, helps determine the circle's center and radius.

This understanding allows us to accurately interpret the given circle's equation while simplifying calculations.

Equation Matching

Equation matching is the process of comparing a given equation with standard forms to identify the type of object it represents. When given options, identifying the correct one is based on critical inspection of the form.

• In the exercise, the equation \(x^2 + y^2 = 5\) is identified as a circle centered at \((0,0)\) with radius \(\sqrt{5}\).
• Among choices like parabola or other conic sections, recognizing it as a circle is essential for matching with the options.

Practice in matching will not only sharpen recognition skills but also deepen understanding of mathematical properties in conic sections.

Center and Radius

The center and radius are pivotal features of a circle equation, offering vital insight into the circle's properties and location on a plane.

• The center \((h, k)\) of a circle is derived from the standard form, reflecting its position relative to the origin.
• For \(x^2 + y^2 = r^2\), the center is \((0, 0)\), indicating the circle is based at the origin.
• The radius \(r\) is calculated as the square root of the constant on the equation's right-hand side, such as \(\sqrt{5}\) for \(x^2 + y^2 = 5\).

Being proficient in extracting these details from a standard equation enhances problem-solving skills in geometry and algebra.