Frequently Asked Questions about Circles

Oct 13, 2025 by windailys.com 85 views

Which Equation Represents a Circle that Contains the Point and Has a Center at a Given Location?

In mathematics, a circle is a set of points that are all equidistant from a central point, known as the center. The distance between the center and any point on the circle is called the radius. In this article, we will explore how to find the equation of a circle that contains a given point and has a center at a specific location.

We are given a point ${$ (-5, -3)$}$ and a center ${$ (-2, 1)$}$. Our goal is to find the equation of the circle that contains the given point and has the center at the specified location. To do this, we need to find the radius of the circle using the distance formula.

The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:

${$ \sqrt{\left(x_2-x_1\right)^{2+\left(y_2-y_1\right)}2}$}$

where ${$ x_1, y_1 $}$ and ${$ x_2, y_2 $}$ are the coordinates of the two points.

To find the radius of the circle, we can use the distance formula to calculate the distance between the center and the given point. Let's substitute the coordinates of the center and the given point into the distance formula:

${$ \sqrt{\left(-2-(-5)\right)^{2+\left(1-(-3)\right)}2}$}$

Simplifying the expression, we get:

${$ \sqrt{\left(3\right)^{2+\left(4\right)}2}$}$

${$ \sqrt{9+16}$}$

${$ \sqrt{25}$}$

${ $5\$}$

Therefore, the radius of the circle is 5 units.

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 $}$ are the coordinates of any point on the circle.

Now that we have found the radius, we can substitute the values into the equation of a circle:

${$ (x-(-2))^2+(y-1)2=5^2$}$

Simplifying the expression, we get:

${$ (x+2)^2+(y-1)2=25$}$

Therefore, the equation of the circle that contains the point ${$ (-5, -3)$}$ and has a center at ${$ (-2, 1)$}$ is:

${$ (x+2)^2+(y-1)2=25$}$

In this article, we have learned how to find the equation of a circle that contains a given point and has a center at a specific location. We used the distance formula to find the radius of the circle and then substituted the values into the equation of a circle. The equation of the circle is ${$ (x+2)^2+(y-1)2=25$}$. This equation represents a circle that contains the point ${$ (-5, -3)$}$ and has a center at ${$ (-2, 1)$}$.
Frequently Asked Questions about Circles

In our previous article, we explored how to find the equation of a circle that contains a given point and has a center at a specific location. In this article, we will answer some frequently asked questions about circles to help you better understand this topic.

A: The center of a circle is the point that is equidistant from all points on the circle. It is the point around which the circle is centered.

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is a measure of the size of the circle.

A: To find the equation of a circle, you need to know the center and the radius of the circle. You can use the distance formula to find the radius, and then substitute the values into the equation of a circle.

A: The equation of a circle with center ${$ h, k $}$ and radius $r$ is given by:

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

A: To find the radius of a circle using the distance formula, you need to know the coordinates of the center and the given point. You can substitute these values into the distance formula:

${$ \sqrt{\left(x_2-x_1\right)^{2+\left(y_2-y_1\right)}2}$}$

A: The radius of a circle is an important concept in mathematics and real-world applications. It is used to calculate the area and circumference of a circle, and it is also used in geometry and trigonometry.

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

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a point, not a circle.

A: To graph a circle, you need to know the center and the radius of the circle. You can use a graphing calculator or a computer program to graph the circle.

A: The center and the radius of a circle are related in that the radius is the distance from the center to any point on the circle.

In this article, we have answered some frequently asked questions about circles to help you better understand this topic. We have covered topics such as the center and radius of a circle, how to find the equation of a circle, and how to graph a circle. We hope that this article has been helpful in your understanding of circles.