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Equation of Circles

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Equation of Circles

We have seen the equation of a line, by which we can represent that line in the Cartesian form. This linear equation is really helpful to find different properties of lines. Similarly, is it possible to provide some equation form for a circle to determine its properties in a Cartesian plane? Well, the answer is YES! Circles can also be represented in the different forms of equations.

This article will cover different forms of equations of a circle in different planes.

Standard form of Equation of a Circle with center and radius

Let us recall the definition of the circle before proceeding with its equation form.

A circle is a two-dimensional closed curve consisting of a set of points at an equal distance from a given point.

The standard form of the equation of a circle provides information about the radius and the center of the circle. We consider any point on the circle with the center point as The distance between these two points is the radius r for that circle.

Then the standard form of the equation for a circle can be written in the following way:

The above form is particularly useful when the coordinates of the center are given straightaway.

Equation of circles, Circle with center and radius, StudySmarterCircle with center and radius, StudySmarter Originals

The standard form of the equation is derived using the distance formula. Considering the distance from the center to any point of the circle, we can represent it as,

Here the distance is actually the radius of the circle, so we equate it with r. Then squaring both sides of the equation, we get the standard form.

Notice the equation contains x-h and y-k, hence an equation with x-3 would shift the graph towards the right by 3. Similarly, an equation of y-3 would shift the graph up by 3.

Find the radius of the circle .

Solution:

From the equation, we can see that 9 corresponds to the term. Hence:

Since the radius of the circle must be positive, we get.

General form of the Equation of Circle

Suppose we are given an equation where all the terms of the equation are expanded and h, k cannot be deduced straightaway. In that case, we further build upon the obtained equation of a circle and derive another form of it, which is more general than the above standard equation.

Expanding the previous equation, it is reduced to:

which can be rearranged as a standard quadratic with squared terms first, followed by the linear terms and then the constant:

To differentiate and avoid the conflict of constants between this equation and the former one, we introduce a set of new constants: to simplify the constant term.

After making these substitutions, we have the following equation of a circle in general form:

The radius of the circle is now given by:

Note that the condition should be fulfilled; otherwise, the radius will not be a positive real number and the circle won’t exist.

One can make little checks after solving an example, just to ensure that the answer makes sense, such as:

  1. The coefficient of x2 and y2 should always be equal, if not, then the equation does not describe a circle.

  2. The inequality is satisfied (otherwise, the radius is a complex number, which it cannot be).

If one of these conditions is not met, then it does not represent a circle.

Equation of Circle in complex form

The equation of a circle in complex form is expressed as below:

Where r is the circle's radius and z0 is the vector shifting to the circle's center.

Since is in the form of , where a and b are constants, the circle shifts horizontally by a factor of a and vertically by a factor of b.

Equation of circles, Circle in complex form, StudySmarterCircle in complex form, StudySmarter Originals

The position of points of the circle is determined with the following conditions:

  1. The points interior of the circle are represented by

  2. The points exterior of the circle are represented by

  3. The circle with the center at origin is represented as , where .

Find the radius and coordinates of the center of the following complex circle:

Solution:

Before we can find these values, we must simplify this equation into the form above. Since ,

Then compare it with the equation of the circle in the complex form we get,

Hence, the coordinates of the circle's center and its radius are (1, 2) and 4 respectively.

Equation of circle formula in Parametric form

The parametric equation of a circle formula by taking a general point (x,y) which makes an angle with the center is given as:

Where r is the radius of the circle. Note that this equation is for the circle with origin as its center.

We consider the coordinates (h,k) as the center of the circle when the center of the circle is not the origin. Then the parametric equation is given by:

We just add the coordinates to x and y as the circle is just shifted from the origin to another location.

Equation of circle examples

Let us take a look at some equation of circle examples.

A circle with center (3,4) is given with the point (2,6) on it. Then find the equation of the given circle.

Solution:

Here the center of the circle is And a point on the circle is

Then the standard equation of the circle is given by

So, by substituting the center coordinates, we get

Now we calculate the radius for this circle by substituting all the given values.

Hence, the equation of the circle with center (3,4) is

Show that is the equation of circle for parametric equations

Solution:

We are given two parametric equations

Then squaring and adding both the equations, we get

What is the center and radius for the equation of a circle

Solution:

The equation of a circle looks similar to the general form of the equation of the circle . So by comparing both the equations we obtain:

From this, we can determine the values of a,b, and c.

Now we know that,

Hence, the center of circle and radius of circle

Equation of Circles - Key takeaways

  • The standard formula for a circle is
  • The center of such a circle has coordinates and a radius r.
  • The general form of the equation of the circle is
  • The complex form of a circle is
  • The parametric equation is for the circle with origin as its center is
  • The parametric equation is for the circle with origin as not its center is

Frequently Asked Questions about Equation of Circles

The equation of a circle is (x-h)2+(y-k)2=r2, which is considered as the standard form.

The radius of a circle can be determined by using the formula r=sqrt(a2+b2-c) from the general equation of a circle.

(x+2)2+(y-3)2=9 is an example of an equation of a circle.

The equation of a circle in complex form is |z-z0|=r.

The equation of a circle in standard form can be determined by applying the distance formula between the point on the circle and its center.

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