Drag Each Equation To The Correct Location In The Table. Not All Equations Will Be Used.Place The Equations That Represent Circles With The Smallest And The Largest Radius Into The Table.1. $2x^2 + 2y^2 + 16x - 4y + 30 = 0$2. $4x^2 + 4y^2

Introduction

In mathematics, equations can be used to represent various geometric shapes, including circles. A circle is defined as the set of all points in a plane that are a fixed distance, called the radius, from a given point, called the center. In this article, we will explore how to identify circles in mathematical equations and determine the smallest and largest radius among them.

Understanding Circle Equations

A circle equation is typically in the form of $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. However, not all circle equations are in this standard form. Some equations may be in the form of $Ax^2 + Ay^2 + Bx + Cy + D = 0$, where $A$, $B$, $C$, and $D$ are constants.

Analyzing the Given Equations

We are given two equations to analyze:

1. $2x^2 + 2y^2 + 16x - 4y + 30 = 0$
2. $4x^2 + 4y^2$

To determine if these equations represent circles, we need to rewrite them in the standard form of a circle equation.

Equation 1: $2x^2 + 2y^2 + 16x - 4y + 30 = 0$

To rewrite this equation in the standard form, we need to complete the square for both the $x$ and $y$ terms.

import sympy as sp
x, y = sp.symbols('x y')

eq1 = 2x**2 + 2y**2 + 16x - 4y + 30

eq1 = sp.expand(eq1)
eq1 = sp.factor(eq1)
print(eq1)

After completing the square, we get:

$2(x + 4)^2 + 2(y - 1)^2 = 2$

Now, we can see that this equation represents a circle with center $(-4, 1)$ and radius $\sqrt{2}$.

Equation 2: $4x^2 + 4y^2$

This equation is already in the standard form of a circle equation, with center $(0, 0)$ and radius $2$.

Determining the Smallest and Largest Radius

Now that we have identified the circles represented by the given equations, we can determine the smallest and largest radius among them.

The smallest radius is $\sqrt{2}$, which corresponds to the circle represented by Equation 1.

The largest radius is $2$, which corresponds to the circle represented by Equation 2.

Conclusion

In this article, we have explored how to identify circles in mathematical equations and determine the smallest and largest radius among them. We have analyzed two given equations and rewritten them in the standard form of a circle equation. We have also determined the smallest and largest radius among the circles represented by the given equations.

Table of Circle Equations

Introduction

In our previous article, we explored how to identify circles in mathematical equations and determine the smallest and largest radius among them. In this article, we will answer some frequently asked questions about circle equations.

Q: What is the standard form of a circle equation?

A: The standard form of a circle equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.

Q: How do I rewrite a circle equation in the standard form?

A: To rewrite a circle equation in the standard form, you need to complete the square for both the $x$ and $y$ terms. This involves adding and subtracting constants to create perfect square trinomials.

Q: What is the center of a circle?

A: The center of a circle is the point in the plane that is equidistant from all points on the circle. It is represented by the coordinates $(h, k)$ in the standard form of a circle equation.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is represented by the constant $r$ in the standard form of a circle equation.

Q: How do I determine the smallest and largest radius among multiple circles?

A: To determine the smallest and largest radius among multiple circles, you need to compare the values of the radius for each circle. The smallest radius is the smallest value, and the largest radius is the largest value.

Q: Can a circle equation have a negative radius?

A: No, a circle equation cannot have a negative radius. The radius of a circle is always a non-negative value.

Q: Can a circle equation have a zero radius?

A: Yes, a circle equation can have a zero radius. This represents a degenerate circle, which is a single point.

Q: How do I graph a circle equation?

A: To graph a circle equation, you need to plot the center of the circle and then draw a circle with the given radius. You can use a graphing calculator or software to help you graph the circle.

Q: Can I use a circle equation to find the area of a circle?

A: Yes, you can use a circle equation to find the area of a circle. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle.

Q: Can I use a circle equation to find the circumference of a circle?

A: Yes, you can use a circle equation to find the circumference of a circle. The circumference of a circle is given by the formula $C = 2\pi r$, where $r$ is the radius of the circle.

Conclusion

In this article, we have answered some frequently asked questions about circle equations. We have covered topics such as the standard form of a circle equation, completing the square, and determining the smallest and largest radius among multiple circles. We hope that this article has been helpful in clarifying any questions you may have had about circle equations.