General Form of the Equation of a Circle

We will discuss about the general form of the equation of a circle.

Prove that the equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 always represents a circle whose centre is (-g, -f) and radius = \(\sqrt{g^{2} + f^{2} - c}\), where g, f and c are three constants

 Conversely, a quadratic equation in x and y of the form x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 always represents the equation of a circle.

We know that the equation of the circle having centre at (h, k) and radius = r units is

(x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\)

x\(^{2}\) + y\(^{2}\) - 2hx - 2hy + h\(^{2}\) + k\(^{2}\) = r\(^{2}\)

x\(^{2}\) + y\(^{2}\) - 2hx - 2hy + h\(^{2}\) + k\(^{2}\) - r\(^{2}\) = 0

Compare the above equation x\(^{2}\) + y\(^{2}\) - 2hx - 2hy + h\(^{2}\) + k\(^{2}\) - r\(^{2}\) = 0 with x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 we get, h = -g, k = -f and h\(^{2}\) + k\(^{2}\) - r\(^{2}\) = c

Therefore the equation of any circle can be expressed in the form x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0.

Again, x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0

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

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

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

This is of the form (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\) which represents a circle having centre at (- g, -f) and radius \(\sqrt{g^{2} + f^{2} - c}\).

Hence the given equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 represents a circle whose centre is (-g, -f) i.e, (-\(\frac{1}{2}\) coefficient of x, -\(\frac{1}{2}\) coefficient of y) and radius = \(\sqrt{g^{2} + f^{2} - c}\) = \(\sqrt{(\frac{1}{2}\textrm{coefficient of x})^{2} + (\frac{1}{2}\textrm{coefficient of y})^{2} - \textrm{constant term}}\)


Note:

(i) The equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 represents a circle of radius = \(\sqrt{g^{2} + f^{2} - c}\).

(ii) If g\(^{2}\) + f\(^{2}\) - c > 0, then the radius of the circle is real and hence the equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 represents a real circle.

(iii) If g\(^{2}\) + f\(^{2}\) - c = 0 then the radius of the circle becomes zero. In this case, the circle reduces to the point (-g, -f). Such a circle is known as a point circle. In other words, the equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 represents a point circle.

(iv) If g\(^{2}\) + f\(^{2}\) - c < 0, the radius of the circle \(\sqrt{g^{2} + f^{2} - c}\) becomes imaginary but the circle is real. Such a circle is called an imaginary circle. In other words, equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 does not represent any real circle as it is not possible to draw such a circle.




11 and 12 Grade Math 

From General Form of the Equation of a Circle to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.