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.
● The Circle
11 and 12 Grade Math
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