Equations (Part 1)


Welcome to the second lesson on circles. This lesson will talk about various forms of equations to a circle in the Cartesian plane. So let’s begin!

(1) Equation of a Circle with center C (x, y) and radius r

Let P(x, y) be any point on the circle. Then, following the definition of the circle, we have CP = r

Circle

Using distance formula, we get the required equation as \( \sqrt{(x-x_1)^2+(y-y_1)^2}=r\), or (x – x1)2 + (y – y­1)2 = r2­­

(2) General Equation

If you expand the equation obtained just now, you’ll get something like this: x2 + y2 – 2x1x – 2y­1y + x12 + y12 – r2­­ = 0

The equation has two terms of second degree: x2 and y, which have the same coefficient (1 in this case); two terms of first degree: – 2x1x and – 2y­1y; and one constant term.

Now consider this equation: x2 + y2 + 2gx + 2fy + c = 0, which looks somewhat like the one above (i.e. the number and types of second degree, first degree and constant terms)

Will this represent a circle? (because it looks like the equation of a circle?). We did something like this when we were talking about the general equation of a straight line.

Turns out it will. We just have to change its form to the one like we just derived.

I’ve added and subtracted a few terms to the equation, and rearranged the terms, so that it looks like this: x2 + 2gx + g2 + y2 + 2fy + f2 = g2 + f2 – c

The equation can now be written as (x + g)2 + (y + f)2 = (\(\sqrt{g^2+f^2-c}\))2 which will look exactly like (x – x1)2 + (y – y­1)2 = r2­­, on substituting g = – x1, f = – y1 and g2 + f2 – c = r2

Now since (x – x1)2 + (y – y­1)2 = r2 represents a circle whose center is (x1, y1) and radius r, therefore, x2 + y2 + 2gx + 2fy + c = 0 will also represent a circle whose center is (–g, –f) and radius equals \(\sqrt{g^2+f^2-c}\) (using the substitutions we made)

The only thing we need to ensure that g2 + f2 – c must be positive for the radius to be defined. In case 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)

To summarize:

(1) Equation of the circle whose center is (x1, y1) and radius is r is given by (x – x1)2 + (y – y­1)2 = r2­­

(2) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0, whose 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.

Leave a comment