Which Equation Represents A Circle That Contains The Point { (-2, 8)$}$ And Has A Center At { (4, 0)$}$?Use The Distance Formula: { \sqrt{(x_2-x_1) 2+(y_2-y_1) 2}$}$.A. { (x-4) 2+y 2=100$}$ B.

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

In mathematics, a circle is a set of points that are equidistant from a central point known as the center. The equation of a circle can be represented in various forms, including the standard form, which is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius. In this article, we will use the distance formula to find the equation of a circle that contains the point (−2,8){(-2,8)} and has a center at (4,0){(4,0)}.

The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by (x2−x1)2+(y2−y1)2{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}. This formula can be used to find the distance between any two points in a coordinate plane.

To find the equation of the circle, we need to find the radius of the circle. The radius is the distance between the center of the circle and any point on the circle. We can use the distance formula to find the radius of the circle.

Let (x1,y1)=(4,0){(x_1,y_1)=(4,0)} be the center of the circle and (x2,y2)=(−2,8){(x_2,y_2)=(-2,8)} be the point on the circle. Using the distance formula, we can calculate the distance between the center and the point:

(x2−x1)2+(y2−y1)2=(−2−4)2+(8−0)2=(−6)2+82=36+64=100=10{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(-2-4)^2+(8-0)^2}=\sqrt{(-6)^2+8^2}=\sqrt{36+64}=\sqrt{100}=10}

Therefore, the radius of the circle is 10{10}.

Now that we have found the radius of the circle, we can find the equation of the circle. The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.

Substituting the values of the center and the radius, we get:

(x−4)2+(y−0)2=102{(x-4)^2+(y-0)^2=10^2}

Simplifying the equation, we get:

(x−4)2+y2=100{(x-4)^2+y^2=100}

Therefore, the equation of the circle that contains the point (−2,8){(-2,8)} and has a center at (4,0){(4,0)} is (x−4)2+y2=100{(x-4)^2+y^2=100}.

In this article, we used the distance formula to find the equation of a circle that contains the point (−2,8){(-2,8)} and has a center at (4,0){(4,0)}. We found the radius of the circle by calculating the distance between the center and the point, and then used the equation of a circle to find the equation of the circle. The equation of the circle is (x−4)2+y2=100{(x-4)^2+y^2=100}.

The correct answer is:

A. (x−4)2+y2={(x-4)^2+y^2=}

This problem requires the use of the distance formula and the equation of a circle. The distance formula is used to find the radius of the circle, and the equation of a circle is used to find the equation of the circle. This problem is a good example of how to use the distance formula and the equation of a circle to solve a problem in mathematics.

  • Equation of a Circle: The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.
  • Distance Formula: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by (x2−x1)2+(y2−y1)2{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}.
  • Geometry: Geometry is the branch of mathematics that deals with the study of shapes and sizes of objects.
  • Mathematics: Mathematics is the study of numbers, quantities, and shapes.
  • Geometry: Geometry is the branch of mathematics that deals with the study of shapes and sizes of objects.
  • Equation of a Circle: The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.
    Q&A: Which Equation Represents a Circle that Contains the Point and Has a Center at?

In our previous article, we discussed how to find the equation of a circle that contains the point (−2,8){(-2,8)} and has a center at (4,0){(4,0)}. We used the distance formula to find the radius of the circle and then used the equation of a circle to find the equation of the circle. In this article, we will answer some frequently asked questions related to the topic.

A: The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.

A: To find the radius of a circle, you can use the distance formula. The distance formula is given by (x2−x1)2+(y2−y1)2{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}. You can use this formula to find the distance between the center of the circle and any point on the circle.

A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by (x2−x1)2+(y2−y1)2{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}.

A: To find the equation of a circle that contains a point and has a center at a given point, you can follow these steps:

  1. Find the distance between the center of the circle and the point using the distance formula.
  2. Use the equation of a circle to find the equation of the circle.

A: The equation of the circle that contains the point (−2,8){(-2,8)} and has a center at (4,0){(4,0)} is (x−4)2+y2=100{(x-4)^2+y^2=100}.

A: You can use the equation of a circle to solve problems by substituting the values of the center and the radius into the equation. You can then use the equation to find the equation of the circle.

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

  • Geometry: The equation of a circle is used to study the properties of circles and their relationships with other shapes.
  • Trigonometry: The equation of a circle is used to solve problems involving right triangles and trigonometric functions.
  • Physics: The equation of a circle is used to model the motion of objects in circular paths.

In this article, we answered some frequently asked questions related to the equation of a circle. We discussed how to find the of a circle, how to use the distance formula, and how to use the equation of a circle to solve problems. We also discussed some real-world applications of the equation of a circle.

  • Mathematics: Mathematics is the study of numbers, quantities, and shapes.
  • Geometry: Geometry is the branch of mathematics that deals with the study of shapes and sizes of objects.
  • Equation of a Circle: The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.
  • Equation of a Circle: The equation of a circle is given by (x−h)2+(y−k)2=r2{(x-h)^2+(y-k)^2=r^2}, where (h,k){(h,k)} is the center of the circle and r{r} is the radius.
  • Distance Formula: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by (x2−x1)2+(y2−y1)2{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}.
  • Geometry: Geometry is the branch of mathematics that deals with the study of shapes and sizes of objects.