Write An Equation In Standard Form Of The Circle With The Given Properties.Center At { (-15, 0)$}$; { R = \sqrt{13}$}$.

Mar 10, 2025 by ADMIN 120 views

**Equation of a Circle in Standard Form**

Introduction

In mathematics, a circle is a set of points that are equidistant from a central point known as the center. The standard form of the equation of a circle is given by ${(x - h)^2 + (y - k)^2 = r^2}$ , where ${(h, k)}$ represents the coordinates of the center of the circle and ${r}$ is the radius. In this article, we will derive the equation of a circle in standard form given its center and radius.

Given Properties

The center of the circle is given as ${(-15, 0)}$ and the radius is given as ${r = \sqrt{13}}$ .

Derivation of the Equation

To derive the equation of the circle in standard form, we will use the formula ${(x - h)^2 + (y - k)^2 = r^2}$ . Since the center of the circle is ${(-15, 0)}$ , we can substitute these values for ${h}$ and ${k}$ in the formula.

Substituting the Center Coordinates

Substituting the center coordinates ${(-15, 0)}$ for ${h}$ and ${k}$ in the formula, we get:

${(x - (-15))^2 + (y - 0)^2 = r^2}$

Simplifying the equation, we get:

${(x + 15)^2 + y^2 = r^2}$

Substituting the Radius

Substituting the radius ${r = \sqrt{13}}$ for ${r^2}$ in the equation, we get:

${(x + 15)^2 + y^2 = (\sqrt{13})^2}$

Simplifying the equation, we get:

${(x + 15)^2 + y^2 = 13}$

Standard Form of the Equation

The standard form of the equation of the circle is:

${(x + 15)^2 + y^2 = 13}$

This is the equation of the circle in standard form with the given properties.

Conclusion

In this article, we derived the equation of a circle in standard form given its center and radius. We used the formula ${(x - h)^2 + (y - k)^2 = r^2}$ and substituted the given values for ${h}$ , ${k}$ , and ${r}$ to obtain the equation of the circle in standard form. The equation of the circle in standard form is ${(x + 15)^2 + y^2 = 13}$ .

Properties of the Circle

The circle has the following properties:

Center: The center of the circle is ${(-15, 0)}$ .
Radius: The radius of the circle is ${\sqrt{13}}$ .
Equation: The equation of the circle in standard form is ${(x + 15)^2 + y^2 = 13}$ .

Graph of the Circle

The graph of the circle is a set of points that are equidistant from the center ${(-15, 0)}$ . The circle is centered at ${(-15, 0)}$ and has a radius of ${\sqrt{13}}$ .

Real-World Applications

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

Geometry: The equation of a circle in standard form is used to describe the shape and size of a circle.
Physics: The equation of a circle in standard form is used to describe the motion of objects in circular motion.
Engineering: The equation of a circle in standard form is used to design and analyze circular structures such as bridges and tunnels.

Summary

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 ${(h, k)}$ represents the coordinates of the center of the circle and ${r}$ is the radius.

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

A: To find the equation of a circle in standard form, you need to know the coordinates of the center of the circle and the radius. You can use the formula ${(x - h)^2 + (y - k)^2 = r^2}$ and substitute the given values for ${h}$ , ${k}$ , and ${r}$ .

Q: What is the center of the circle in the given equation?

A: The center of the circle in the given equation is ${(-15, 0)}$ .

Q: What is the radius of the circle in the given equation?

A: The radius of the circle in the given equation is ${\sqrt{13}}$ .

Q: How do I graph a circle?

A: To graph a circle, you need to know the center and radius of the circle. You can use the equation of the circle in standard form to graph the circle.

Q: What are some real-world applications of the equation of a circle?

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

Geometry: The equation of a circle is used to describe the shape and size of a circle.
Physics: The equation of a circle is used to describe the motion of objects in circular motion.
Engineering: The equation of a circle is used to design and analyze circular structures such as bridges and tunnels.

Q: How do I find the distance between two points on a circle?

A: To find the distance between two points on a circle, you can use the equation of the circle in standard form and the distance formula.

Q: What is the equation of a circle with a center at ${(0, 0)}$ and a radius of ${5}$ ?

A: The equation of a circle with a center at ${(0, 0)}$ and a radius of ${5}$ is ${x^2 + y^2 = 25}$ .

Q: What is the equation of a circle with a center at ${(3, 4)}$ and a radius of ${2}$ ?

A: The equation of a circle with a center at ${(3, 4)}$ and a radius of ${2}$ is ${(x - 3)^2 + (y - 4)^2 = 4}$ .

Q: How do I find the equation of a circle given three points on the circle?

A: To find the equation of a circle given three points on the circle, you can use the following steps:

Find the center of the circle by finding the intersection of the perpendicular bisectors of the three points.
Find the radius of the circle by finding the distance from the center to one of the points.
Use the equation of the circle in standard form and substitute the values for the center and radius.

Q: What is the equation of a circle with a center at ${(-2, 1)}$ and a radius of ${3}$ ?

A: The equation of a circle with a center at ${(-2, 1)}$ and a radius of ${3}$ is ${(x + 2)^2 + (y - 1)^2 = 9}$ .

Q: How do I find the area of a circle?

A: To find the area of a circle, you can use the formula ${A = \pi r^2}$ , where ${r}$ is the radius of the circle.

Q: What is the area of a circle with a radius of ${4}$ ?

A: The area of a circle with a radius of ${4}$ is ${A = \pi (4)^2 = 16\pi}$ .

Q: How do I find the circumference of a circle?

A: To find the circumference of a circle, you can use the formula ${C = 2\pi r}$ , where ${r}$ is the radius of the circle.

Q: What is the circumference of a circle with a radius of ${5}$ ?

A: The circumference of a circle with a radius of ${5}$ is ${C = 2\pi (5) = 10\pi}$ .