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
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