An Equation Of A Circle Is Given By $(x+3)^2 + (y-9)^2 = 5^2$. Apply The Distributive Property To The Squared Binomials And Rearrange The Equation So That One Side Is 0.

Feb 27, 2025 by ADMIN 170 views

**An Equation of a Circle: Simplifying and Rearranging**

Understanding the Given Equation

The equation of a circle is given by $(x+3)^2 + (y-9)^2 = 5^2$ . This equation represents a circle with a center at $(-3, 9)$ and a radius of $5$ . To simplify and rearrange this equation, we will apply the distributive property to the squared binomials and then rearrange the terms to have one side equal to zero.

Applying the Distributive Property

To apply the distributive property, we will expand the squared binomials using the formula $(a+b)^2 = a^2 + 2ab + b^2$ . For the first squared binomial, $(x+3)^2$ , we have:

$(x+3)^2 = x^2 + 2(x)(3) + 3^2$ $(x+3)^2 = x^2 + 6x + 9$

Similarly, for the second squared binomial, $(y-9)^2$ , we have:

$(y-9)^2 = y^2 - 2(y)(9) + 9^2$ $(y-9)^2 = y^2 - 18y + 81$

Rearranging the Equation

Now that we have expanded the squared binomials, we can substitute these expressions back into the original equation:

$x^2 + 6x + 9 + y^2 - 18y + 81 = 5^2$

We can simplify this equation by combining like terms:

$x^2 + 6x + y^2 - 18y + 90 = 25$

Next, we can rearrange the equation to have one side equal to zero:

$x^2 + 6x + y^2 - 18y + 65 = 0$

Conclusion

In this article, we simplified and rearranged the equation of a circle using the distributive property. We expanded the squared binomials and then substituted these expressions back into the original equation. Finally, we rearranged the equation to have one side equal to zero. This process is an important step in understanding the properties of circles and how to work with their equations.

Properties of Circles

Circles are a fundamental concept in mathematics, and their equations are used to describe their properties. The equation of a circle is given by $(x-h)^2 + (y-k)^2 = r^2$ , where $(h, k)$ is the center of the circle and $r$ is the radius. The equation we simplified and rearranged in this article is a specific example of this general form.

Real-World Applications

The equation of a circle has many real-world applications, including:

Geometry: The equation of a circle is used to describe the properties of circles and their relationships to other geometric shapes.
Physics: The equation of a circle is used to describe the motion of objects in circular paths.
Engineering: The equation of a circle is used to design and optimize circular structures, such as bridges and tunnels.

Future Directions

In this article, we simplified and rearranged the equation of a circle using the distributive property. However, there are many other ways to simplify and rearrange this equation, and exploring these different approaches can provide valuable insights into the properties of circles and their equations.

Simplifying the Equation Further

One way to simplify the equation further is to complete the square. This involves adding and subtracting a constant term to create a perfect square trinomial. For example, we can add and subtract $9$ to the first term:

$x^2 + 6x + 9 + y^2 - 18y + 81 = 25$ $x^2 + 6x + 9 + y^2 - 18y + 81 - 9 + 9 = 25 - 9$ $(x+3)^2 + y^2 - 18y + 81 = 16$

This simplified equation has a more compact form and can be used to describe the properties of the circle.

Conclusion

In this article, we simplified and rearranged the equation of a circle using the distributive property and completing the square. We explored the properties of circles and their equations, and we discussed the real-world applications of the equation of a circle. We also touched on future directions for simplifying and rearranging the equation of a circle.

Understanding the Equation of a Circle

The equation of a circle is a fundamental concept in mathematics that describes the properties of circles and their relationships to other geometric shapes. In this article, we will answer some of the most frequently asked questions about the equation of a circle.

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

A: The general form of the equation of a circle 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 find the center and radius of a circle from its equation?

A: To find the center and radius of a circle from its equation, you need to rewrite the equation in the general form $(x-h)^2 + (y-k)^2 = r^2$ . The values of $h$ and $k$ will give you the coordinates of the center, and the value of $r$ will give you the radius.

Q: What is the difference between the equation of a circle and the equation of an ellipse?

A: The equation of a circle is a special case of the equation of an ellipse, where the two axes are equal in length. The equation of an ellipse is more general and can be written as $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

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

A: Yes, you can use the equation of a circle 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. You can find the radius of the circle from its equation and then use this formula to find the area.

Q: How do I graph the equation of a circle?

A: To graph the equation of a circle, you need to find the center and radius of the circle from its equation. Then, you can use a graphing calculator or a computer program to plot the circle. You can also use a compass and a straightedge to draw the circle by hand.

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

A: Yes, you can use the equation of a circle 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. You can find the radius of the circle from its equation and then use this formula to find the circumference.

Q: What is the relationship between the equation of a circle and the equation of a sphere?

A: The equation of a sphere is a three-dimensional extension of the equation of a circle. The equation of a sphere is given by $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$ , where $(h, k, l)$ is the center of the sphere and $r$ is the radius.

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

A: Yes, you can use the equation of a circle to find the volume of a sphere. The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere. You can find the radius of the sphere from its equation and then use this formula to find the volume.

Conclusion

In this article, we answered some of the most frequently asked questions about the equation of a circle. We covered topics such as the general form of the equation of a circle, finding the center and radius of a circle, and using the equation of a circle to find the area, circumference, and volume of a circle. We also discussed the relationship between the equation of a circle and the equation of a sphere.