What is the equation of a circle? And how does it help to identify or describe the location of a circular object? Can you determine the circular objects in your surroundings?

The equation of a circle does not represent the area of a circle equation. Instead, it provides an algebraic way to describe the circle’s position or the family of circles in a Cartesian plane. It contains only the coordinates of the center, a fixed point inside the circle, and the radius, which is the distance from the center to the boundary of the circle. Moreover, the circle equation denotes all the points lying on its circumference. So what is the equation of a circle?

**Equation of a Circle**

After learning the equation of a circle, let us learn how to derive the standard equation of a circle. The circle’s center coordinates are denoted by (a, b) as shown in the figure, and the radius is represented by r and (h, k) are the arbitrary points located on the circle’s circumference.

**Derivation of Circle Equation**

The distance between the arbitrary point and the center of the circle is equal to the radius of the circle. From the distance formula, we get,

√(h-a)²+(k-b)²=r

On squaring both the sides, we get,

(h – a)²+ (k – b)² = r², which is the standard equation of the circle.

The general notation for representing the circle uses x and y as arbitrary points. Therefore, the equation of the circle becomes (x – a)² + (x – y)² = r².

However, this representation of the circle equation is not the only one to be used across the world. There are several other forms of notation as well.

**Various Forms of Representing the Equation of a Circle**

Do you know the circle equation can be represented in multiple ways? The equation of a circle can be expressed in various forms depending on the circle’s position in the Cartesian plane. The multiple forms of representing the circle are:

**General form**

x^{2} + y^{2} + 2gx + 2fy + c = 0 is the general form of the equation of the circle. x and y are the arbitrary points on the circumference of the circle, and g, f, and c are the constants. This general form of the equation is used to locate the radius and the coordinates of the center of the circle. Unlike the standard form of the equation of the circle, the general form is difficult to understand and find some meaningful properties of the circle. However, the general form of the equation of the circle is helpful to find the family of circles in the Cartesian plane.

**Standard form**

Most scientists and mathematicians use the standard form equation of a circle because it gives precise information about the center and radius of a circle. Moreover, the standard form equation of a circle is easier to understand and read. The standard form equation of a circle is given by:

(x -x_{1})² + (y – y_{1})² = r², where (x, y) is the arbitrary coordinates on the circumference of the circle, r is the radius of the circle, and (x_{1}, y_{1}) are the coordinates of the center of the circle. The standard form of the equation of the circle is derived from the distance formula.

**Parametric form**

To find the parametric form of the circle equation, deduce the general form of the circle x² + y² + 2hx + 2ky + C = 0. For this, take a general point on the boundary of the circle, for instance, (x, y), and join this general point with the center of the circle (-h. -k). When these points are joined, they form an angle 𝜃. Therefore, the parametric equation of the circle can be written as:

x²+ y²+ 2hx + 2ky + C = 0, where x = -h + r cos𝜃 and y =k + r sin𝜃.

**Polar form**

Polar form representation is similar to the parametric form of the circle equation. The polar form is mostly used to represent the equation of the circle whose center is at the origin. For this, take an arbitrary point A having coordinates (r cos𝜃, r sin𝜃) on the periphery of the circle and a radius r, which is the distance between the random point and the origin. The equation of the circle having radius A and center at the origin will be given by, x²+ y² = A².

Putting the values of x = r cos𝜃 and y = r sin𝜃, we get,

(r cos𝜃)² + (r sin𝜃)² = A²

r²cos²θ + r²sin²θ = A²

r²(cos²θ + sin²θ) = A²

r²(1) = A²(Since, cos²θ + sin²θ = 1 from the trigonometric identities)

r = A

where p is the radius of the circle. Thus, the polar form is used to find the radius of the circle from the standard form of the equation of the circle.

**Steps to Find the Equation of a Circle**

We have seen different ways of representing the equation of a circle depending upon the position of the center of the circle in the Cartesian plane. Therefore, to write the equation of a circle, when the coordinates of the center are given, one can follow these steps:

**Step 1:** Figure out the coordinates of the center of the circle (x_{1}, y_{1}) and the radius of the circle.

**Step 2:** With the help of the equation of the circle formula (x -x_{1})² + (y – y_{1})² = r², allocate the values of the radius and center of the circle.

**Step 3:** After simplifying the equation, the equation of the circle will be obtained.

**Understanding the Equation of a Circle With Examples**

**Example 1: Find the equation of a circle whose center passes through the origin.**

**Solution:** We know, the equation of a circle is given by,

(x – x_{1})² + (y -y_{1})² = r², where x_{1} and y_{1} are the coordinates of the center of the circle and r is the radius.

Since the circle passes through the origin, the values of x_{1} and y_{1} will be zero.

Therefore, the standard equation of the circle passing through the origin will be x² + y² = r².

**Example 2: What will be the general equation of a circle whose radius is 6 units and the center lies on (4, 2)?**

**Solution:** To find the solution to this question, one does not need an equation from a circle calculator. We know, the general equation of a circle is given by, (x – x_{1})² + (y – y_{1})² = r². Putting the value of centers at x_{1} and y_{1} as 4 and 2, along with the radius of 6 units, the equation of the circle becomes

(x – 4)² + (y – 2)² = 6²

x² + 16 – 8x + y² + 4 – 4y = 36

Simplifying the above equation, we get the final equation of circle as,

x² + y² – 8x – 4y = 16.

**Example 3: Find the radius of the standard equation by converting the following standard equation of a circle into the polar form: x² + y²= 25.**

**Solution:** To convert the standard equation of the circle into polar form, substitute the value of x with r cos𝜃 and y with r sin𝜃.

Therefore, the standard equation of the circle now becomes, (r cos𝜃)² + (r sin𝜃)² = 25.

On solving the above equation, we get, r² cos²𝜃 + r² sin²𝜃 = 25

r² (cos²𝜃 + sin²𝜃) = 25. We know, cos²𝜃 + sin²𝜃 = 1.

Substituting this value, we get r²= 25. This gives r = 5 units.

Therefore, the radius of the circle is 5 units.

**Example 4: What is the value of the center of the circle and radius if the standard equation of the circle is (x + 7)² + (y – 9)² = 529?**

**Solution:** To find the value of the center of the circle and radius, compare the given equation of the circle with the standard form, i.e., (x – x_{1})² + (y – y_{1})² = r².

On comparing, we see that the value of

x_{1} = -7, y_{1} = 9 and r = 23.

Therefore, the coordinates of the center of the circle are (-7, 9) and radius = 23.

**Frequently Asked Questions**

**1. How to derive the equation for a circle?**

**Ans: **The circle’s center coordinates are denoted by (a, b) as shown in the figure, and the radius is represented by r and (h, k) are the arbitrary points located on the circle’s circumference. The distance between the arbitrary point and the center of the circle is equal to the radius of the circle. From the distance formula, we get

√(h-a)²+(k-b)²=r

On squaring both sides, we get,

(h – a)²+ (k – b)² = r², which is the standard equation of the circle.

**2. What is the equation for a circle with origin (0,0) as its center?**

**Ans:** Substituting the values for (a,b) we get,

(h – 0)²+ (k – 0)² = r²

h²+ k² = r²

is the equation for a circle with origin as its center where (h, k) are the arbitrary points located on the circle’s circumference

**3. Are There Various Forms for the Equation of a Circle?**

**Ans:** Different forms of circle equations are:

General Form:

x^{2}+y^{2}+2gx+2fy+c=0

Standard Form: (h – a)²+ (k – b)² = r² where (h, k) are the arbitrary points located on the circle’s circumference and (a,b) are the center coordinates.

Polar Form: r=A is the Polar form of a Circle equation.

**4. What is the Polar Form of the Circle Equation?**

**Ans:** Replace the value of x = r cos θ and y = r sin θ, in x2 + y2 = a2.

We get,(r cos θ)^{2}+ (r sin θ)^{2} = a^{2}

r^{2}cos^{2} θ+ r^{2}sin^{2} θ = a^{2}

r^{2}(cos^{2} θ+ sin^{2} θ) = a^{2}

r^{2}(1) = a^{2} (Since, cos²θ + sin²θ = 1 from the trigonometric identities)

r = a

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