Equations (Part 1)

In this lesson, we’ll talk about various forms of equations to a circle on the Cartesian plane. Let’s begin!

Equation of a Circle with Centre (x_1­, y_1­) and Radius r

To find the equation, we’ll pick any point on the circle and establish some relation between it’s coordinates that is always true. This relation will be the required equation.

Let C(x₁, y₁) be the center of the circle. Then, for any point P(x, y) on the circle, we have:

CP = r

Why? Because any point on the circumference is always at a fixed distance from the center of the circle. And this fixed distance is the radius.

Since we’re looking for a relation between the coordinates, we’ll bring in some coordinate geometry formulas. Using the distance formula, we can rewrite the above relation as:

\( \sqrt{(x-x_1)^2+(y-y_1)^2}=r\)

Finally, to make it look nicer, we’ll square both sides to get:

(x – x₁)² + (y – y­₁)² = r^2­­

Since the above relation is true for any point on the circle, it is the required equation of the circle.

General Equation of a Circle

Now, we’ve found the standard equation as well as the one above. But what kind of equation in general would represent a circle?

For example, which of the following equations represent a circle?

x² + y – 2x – 4 = 0

x + y² + 6x – 2y + 5 = 0

3x + 4y = 12

x² + y² – 2x + 4y + 1 = 0

The third equation represents a line, something that we learnt in a previous lesson. What about the other ones?

To answer that, we need to get a hang of the equation of any circle. Just like we did in this lesson, for a line.

So, let’s pick any circle, i.e. a circle with the center as (x₁, y₁) and radius as r, where x₁, y₁, and r (> 0) can have any value. Now, we’ll find it’s equation and look closely at the kind of terms we get.

From the previous section, the equation of this circle will be:

(x – x₁)² + (y – y­₁)² = r^2­­

If we expand the terms on the LHS, we’ll get something like this:

x² + y² – 2x₁x – 2y­₁y + x₁² + y₁² – r^2­­ = 0

The equation has the following types of terms.

Two terms of second degree: x² and y²

Two terms of first degree: –2x₁x and –2y­₁y

One constant term: –r^2­­

Also, the second degree terms have the same coefficient (1 in this case).

Now consider this equation:

x² + y² + 2gx + 2fy + c = 0

This somewhat looks like the previous one (i.e. with respect to the number and types of terms).

Does this represent a circle, because it looks like the equation of a circle?

Turns out it does. We just have to make it look exactly like the one in the previous section. Let’s do that!

x² + y² + 2gx + 2fy + c = 0

⇒ (x² + 2gx) + (y² + 2fy) + c = 0 (rearranging the terms)

⇒ (x² + 2gx + g²) + (y² + 2fy + f²) + c = 0 + g² + f² (adding g² and f² on both sides)

⇒ (x² + 2gx + g²) + (y² + 2fy + f²) = g² + f² – c (subtracting c from both sides)

The equation can now be written as:

(x + g)² + (y + f)² = (\(\sqrt{g^2+f^2-c}\))²

This looks exactly like:

(x – x₁)² + (y – y­₁)² = r^2­­,

where g = –x₁, f = –y₁ and \(\sqrt{g^2+f^2-c}\) = r

Now,

(x – x₁)² + (y – y­₁)² = r² represents a circle whose center is (x₁, y₁) and radius is r.

Therefore,

x² + y² + 2gx + 2fy + c = 0 also represents a circle whose center is (–g, –f) and radius is \(\sqrt{g^2+f^2-c}\)

We’ll that’s it! We just learnt about the general equation of circle.

The only thing we need to ensure that g² + f² – c must be positive for the radius to be defined. If g² + f² – c is zero, the radius of the circle also becomes zero. In this case, the circle is called a point circle (something which you needn’t be worried about right now).

Lesson Summary

1. The equation of the circle whose center is (x₁, y₁) and radius is r is given by (x – x₁)² + (y – y­₁)² = r^2­­.
2. The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the center is (–g, –f) and radius is \(\sqrt{g^2+f^2-c}\).

The next lesson will cover a few examples related to equations of the circle.