Write The Standard Form Of The Equation And The General Form Of The Equation Of The Circle With Radius R R R And Center ( H , K ) {(h, K)} ( H , K ) . Then Graph The Circle. Given: R = 10 R = 10 R = 10 And ( H , K ) = ( − 6 , − 8 ) {(h, K) = (-6, -8)} ( H , K ) = ( − 6 , − 8 ) . The Standard Form Of The Equation Of

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Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will discuss the standard form and general form of the equation of a circle with a given radius and center. We will also provide a step-by-step guide on how to graph the circle.

The Standard Form of the Equation of a Circle

The standard form of the equation of a circle with center ${(h, k)}$ and radius $r$ is given by:

${(x - h)^2 + (y - k)^2 = r^2}$

where $(x, y)$ represents any point on the circle.

The General Form of the Equation of a Circle

The general form of the equation of a circle is given by:

${Ax^2 + Ay^2 + Bx + Cy + D = 0}$

where $A$, $B$, $C$, and $D$ are constants.

Converting the General Form to the Standard Form

To convert the general form to the standard form, we need to complete the square for both the $x$ and $y$ terms.

Completing the Square for the $x$ Term

To complete the square for the $x$ term, we need to add and subtract the square of half the coefficient of the $x$ term.

Completing the Square for the $y$ Term

To complete the square for the $y$ term, we need to add and subtract the square of half the coefficient of the $y$ term.

Writing the Standard Form of the Equation of the Circle

Given that the radius $r = 10$ and the center is $(h, k) = (-6, -8)$, we can write the standard form of the equation of the circle as:

${(x + 6)^2 + (y + 8)^2 = 100}$

Graphing the Circle

To graph the circle, we need to find the center and the radius of the circle.

Finding the Center of the Circle

The center of the circle is given by the coordinates $(h, k) = (-6, -8)$.

Finding the Radius of the Circle

The radius of the circle is given by the value of $r = 10$.

Plotting the Circle

To plot the circle, we need to plot the center and the points on the circle that are a distance of $r$ from the center.

Conclusion

In this article, we discussed the standard form and general form of the equation of a circle with a given radius and center. We also provided a step-by-step guide on how to graph the circle. The standard form of the equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, while the general form is given by $Ax^2 + Ay^2 + Bx + Cy + D = 0$. We also showed how to convert the general form to the standard form by completing the square for both the $x$ and $y$ terms. Finally, we graphed the circle by finding the center and the radius of the circle and the points on the circle that are a distance of $r$ from the center.

Example

Let's consider an example where the radius $r = 5$ and the center is $(h, k) = (2, 3)$. We can write the standard form of the equation of the circle as:

${(x - 2)^2 + (y - 3)^2 = 25}$

To graph the circle, we need to find the center and the radius of the circle. The center of the circle is given by the coordinates $(h, k) = (2, 3)$, while the radius of the circle is given by the value of $r = 5$. We can plot the circle by plotting the center and the points on the circle that are a distance of $r$ from the center.

Applications of Circles

Circles have many applications in mathematics and real-world problems. Some of the applications of circles include:

• Geometry: Circles are used to define the shape and size of geometric figures.
• Trigonometry: Circles are used to define the relationships between angles and sides of triangles.
• Physics: Circles are used to describe the motion of objects in circular paths.
• Engineering: Circles are used to design and build circular structures such as bridges, tunnels, and buildings.

Conclusion

In conclusion, the standard form and general form of the equation of a circle with a given radius and center are essential concepts in mathematics. The standard form of the equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, while the general form is given by $Ax^2 + Ay^2 + Bx + Cy + D = 0$. We also showed how to convert the general form to the standard form by completing the square for both the $x$ and $y$ terms. Finally, we graphed the circle by finding the center and the radius of the circle and plotting the points on the circle that are a distance of $r$ from the center.

Introduction

In our previous article, we discussed the standard form and general form of the equation of a circle with a given radius and center. We also provided a step-by-step guide on how to graph the circle. In this article, we will answer some frequently asked questions (FAQs) about the standard form and general form of the equation of a circle.

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

A: The standard form of the equation of a circle is given by:

${(x - h)^2 + (y - k)^2 = r^2}$

where $(x, y)$ represents any point on the circle, $(h, k)$ is the center of the circle, and $r$ is the radius of the circle.

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

A: The general form of the equation of a circle is given by:

${Ax^2 + Ay^2 + Bx + Cy + D = 0}$

where $A$, $B$, $C$, and $D$ are constants.

Q: How do I convert the general form to the standard form?

A: To convert the general form to the standard form, you need to complete the square for both the $x$ and $y$ terms. This involves adding and subtracting the square of half the coefficient of the $x$ term and the square of half the coefficient of the $y$ term.

Q: What is the center of the circle?

A: The center of the circle is given by the coordinates $(h, k)$.

Q: What is the radius of the circle?

A: The radius of the circle is given by the value of $r$.

Q: How do I graph the circle?

A: To graph the circle, you need to find the center and the radius of the circle. Then, you can plot the center and the points on the circle that are a distance of $r$ from the center.

Q: What are some applications of circles?

A: Circles have many applications in mathematics and real-world problems. Some of the applications of circles include:

• Geometry: Circles are used to define the shape and size of geometric figures.
• Trigonometry: Circles are used to define the relationships between angles and sides of triangles.
• Physics: Circles are used to describe the motion of objects in circular paths.
• Engineering: Circles are used to design and build circular structures such as bridges, tunnels, and buildings.

Q: Can I use the standard form to find the center and radius of the circle?

A: Yes, you can use the standard form to find the center and radius of the circle. The center of the circle is given by the coordinates $(h, k)$, and the radius of the circle is given by the value of $r$.

Q: Can I use the general form to find the center and radius of the circle?

A: Yes, you can use the general form to find the center and radius of the circle. However, you need to complete the square for both the $x$ and $y$ terms to convert the general form to the standard form.

Q: What is the difference between the standard form and the general?

A: The standard form of the equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, while the general form is given by $Ax^2 + Ay^2 + Bx + Cy + D = 0$. The standard form is more convenient to use when graphing the circle, while the general form is more convenient to use when finding the center and radius of the circle.

Conclusion

In conclusion, the standard form and general form of the equation of a circle with a given radius and center are essential concepts in mathematics. We answered some frequently asked questions (FAQs) about the standard form and general form of the equation of a circle. We hope that this article has provided you with a better understanding of the standard form and general form of the equation of a circle.