Equation of a Circle Definitions and Examples
Introduction
The equation of a circle is a mathematical formula used to determine the properties and location of a circle based on certain variables. The equation is also sometimes referred to as the “general form” of the circle equation. A circle is defined as a set of points in a plane that are equidistant from a given point, called the center. The distance between the center and any point on the circle is called the radius. The equation of a circle can be used to determine the radius, center, or circumference of a circle based on certain known values. In this blog post, we will explore the equation of a circle in further detail, including definitions and examples.
What is the Equation of Circle?
A circle is a shape consisting of all points in a plane at a fixed distance, called the radius, from a given point, called the center. Circles are simple closed curves which divide the plane into two regions: an interior and an exterior.
The equation of a circle can be written in several ways. The standard form of the equation of a circle is (x-h)^2 + (y-k)^2 = r^2 where (h, k) is the coordinates of the center of the circle, and r is the radius.
Another way to write the equation of a circle is in terms of x and y intercepts. The x intercepts are where the graph of the circle crosses the x axis, and they occur at x = h +/- r. The y intercepts are where the graph of the circle crosses the y axis, and they occur at y = k +/- r.
The equation of a circle can also be written in terms of its diameter d. The diameter is twice the length of the radius, so it goes through boththe center and one point onthe circumferenceofthe circle. The equation for this case is (x-h)^2 + (y-k)^2 = (d/2)^2 .
Different Forms of Equation of Circle
The standard form of the equation of a circle is:
(x-h)^2 + (y-k)^2 = r^2
where (h, k) are the coordinates of the center of the circle and r is the radius.
However, there are other forms of the equation of a circle that can be useful in certain situations. For example, the parametric form of the equation is:
x = h + r*cos(t)
y = k + r*sin(t)
where t is a parameter. This form is particularly useful when graphing circles on a computer.
Another form of the equation is the polar form:
r = h+ cos (theta – phi)
sin (theta – phi)
where phi is the angle formed by the line segment joining the center of the circle to the point on the circumference and ? is any angle measure. This form is useful when dealing with circles that are not centered at the origin.
Equation of a Circle Formula
The equation of a circle is a mathematical expression that defines the properties of acircle in Euclidean geometry. There are many different ways to write the equation of a circle, but all of them are based on the same fundamental principles. In this article, we’ll introduce the most common form of the equation of a circle and show you how to use it to solve problems.
The most common form of the equation of a circle is:
x2 + y2 = r2
where x and y are the coordinates of any point on the circumference of the circle, and r is the radius of the circle. This equation is known as the standard form of the equation of a circle.
To use this formula, simply substitute the values for x and y into the equation and solve for r. For example, if we know that the coordinates of one point on a circle are (3,4), we can plug those values into our equation and solve for r:
32 + 42 = r2
9 + 16 = r2
25 = r2
5=r
Derivation of Circle Equation
A circle is defined as the set of all points in a plane that are equidistant from a given point, called the center. The distance between any point on the circle and the center is called the radius. The equation of a circle can be derived using the Pythagorean theorem.
The Pythagorean theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In other words, if we have a triangle with sides A, B and C, where C is the hypotenuse, then:
A^2 + B^2 = C^2
We can use this theorem to derive the equation of a circle as follows:
Consider a circle with centre O and radius r (see figure 1). Draw a line from O to any point P on the circumference of the circle. This line forms a right angled triangle with one side being OP (the radius) and two sides being OA and OB (the perpendiculars from O to lines AB and AC respectively). We can therefore write:
r^2 = OA^2 + OB^2 ………………………..(1)
Graphing the Equation of Circle
The equation of a circle is a mathematical formula used to determine the properties of a circle based on its radius, center point, and/or circumference. The following is an example of how to graph the equation of a circle:
1. First, identify the center point and radius of the circle.
2. Next, use the Pythagorean theorem to find the coordinates of the points on the circumference.
3. Finally, connect the dots to graph the equation of the circle.
How to Find Equation of Circle?
To find the equation of a circle, one must first identify the center and radius of the circle. The center is simply the midpoint of the circle, while the radius is the distance from the center to any point on the edge of the circle. Once these two points are identified, one can use them to calculate the equation of the circle using the following formula: (x-h)^2 + (y-k)^2 = r^2. In this formula, h and k represent the coordinates of the center, while r represents the radius.
Converting General Form to Standard Form
A circle can be defined as the set of points in a plane that are equidistant from a given point, called the center. The distance from the center to any point on the circle is called the radius.
The most common form of the equation of a circle is:
x^2 + y^2 = r^2
This is referred to as the “standard form” of the equation. In this form, x and y are the coordinates of any point on the circle, and r is the radius.
To convert an equation from general form to standard form, simply solve for y in terms of x. For example, consider the following equation in general form:
3x^2 + 3y^2 – 6x – 12y + 9 = 0
Solving for y gives us:
y = -(1/3)x^2 – 2x + 3
Converting Standard Form to General Form
To convert equation from standard form to general form, first complete the square on each side of the equal sign. This will put the equation into the form (x-h)^2+(y-k)^2=r^2. Then, identify h and k as the x and y intercepts of the circle, respectively. Finally, r will be the length of a line segment drawn from the origin to any point on the circle.
Conclusion
In conclusion, the equation of a circle is a very important mathematical concept that has a variety of applications. We hope that this article has helped you to better understand what the equation of a circle is and how to use it. If you have any further questions, be sure to consult with a math expert.
