Problem 16 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}=25-y^{2}$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Rewrite the Equation

Start by rewriting the given equation. The equation given is \( x^2 = 25 - y^2 \). We can rearrange it as \( x^2 + y^2 = 25 \). This is the standard form of a conic section equation.

Step 2: Identify Standard Form

The equation \( x^2 + y^2 = 25 \) matches the standard form of a circle. The general form for a circle is \( x^2 + y^2 = r^2 \), where \( r \) is the radius. Here, it is identical to the equation \( (x-0)^2 + (y-0)^2 = 5^2 \).

Step 3: Classify the Conic Section

Since the equation is in the form \( x^2 + y^2 = r^2 \), it represents a circle. A circle is characterized by having both \( x \) and \( y \) squared and added together, with equal coefficients, which is the case here with coefficients of 1 for both variables.

Key Concepts

Circle

A circle is a perfect geometrical shape that is round. Every point on the circle is at an equal distance from a central point known as the center. This distance is called the radius. A circle is characterized by the absence of edges or corners, and it divides the plane into an interior and exterior region.

Important properties of a circle:

• Radius: The constant distance from the center to any point on the circle.
• Diameter: Twice the radius, this is the longest distance across the circle.
• Circumference: The total distance around the circle, which is calculated using the formula \( C = 2\pi r \), where \( r \) is the radius.
• Area: The space contained within the circle, given by \( A = \pi r^2 \).

Understanding these properties aids in solving problems related to circles in mathematics, especially when dealing with equations that describe this distinct shape.

Equation of a Circle

The equation of a circle in mathematics is essential as it precisely defines the circle's location and size on a coordinate plane. The equation can describe where the circle is centered and what its radius is.For a circle centered at the origin

• The equation is \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle.

For a circle centered at a point \((h, k)\)

• The equation becomes \((x-h)^2 + (y-k)^2 = r^2\).

This equation helps us determine different features of a circle, such as its center \((h, k)\) and radius \( r \). When graphing circles, ensuring the equation is in this form helps quickly identify these key characteristics.

Standard Form

Standard form equations are a way of expressing conic sections on a coordinate plane. For circles, this form makes it much easier for students to graph and analyze the properties of the circle.For a circle:

• The standard form is \((x-h)^2 + (y-k)^2 = r^2\).

This formula tells us that the circle is centered at \((h, k)\) with a radius of \( r \). This is crucial for graphing and is widely used in solving geometry problems, allowing one to easily classify the type of conic section based on its equation.

Having equations in standard form allows for easy classification of conic sections by simply comparing their characteristics, such as recognizing that both \( x \) and \( y \) being squared in a specific way indicates a circle.