We will learn how the general equation of second degree represents a circle.
General second degree equation in x and y is
ax\(^{2}\) + 2hxy + by\(^{2}\) + 2gx + 2fy + C = 0, where a, h, b, g, f and c are constants.
If a = b(≠ 0 ) and h = 0, then the above equation becomes
ax\(^{2}\) + ay\(^{2}\) + 2gx + 2fy + c = 0
⇒ x\(^{2}\) + y\(^{2}\) + 2 ∙ \(\frac{g}{a}\) x + 2 ∙ \(\frac{f}{a}\) y + \(\frac{c}{a}\) = 0, (Since, a ≠ 0)
⇒ x\(^{2}\) + 2 ∙ x ∙ \(\frac{g}{a}\) + \(\frac{g^{2}}{a^{2}}\) + y\(^{2}\) + 2.y .\(\frac{f}{a}\) + \(\frac{f^{2}}{a^{2}}\) = \(\frac{g^{2}}{a^{2}}\) + \(\frac{f^{2}}{a^{2}}\)  \(\frac{c}{a}\)
⇒ (x + \(\frac{g}{a}\))\(^{2}\) + (y + \(\frac{f}{a}\))\(^{2}\) = \((\frac{1}{a}\sqrt{g^{2} + f^{2}  ca})^{2}\)
Which represents the
equation of a circle having centre at (\(\frac{g}{a}\), \(\frac{f}{a}\)) and radius = \(\mathrm{\frac{1}{a}\sqrt{g^{2} + f^{2}  ca}}\)
Therefore, the general second degree equation in x and y represents a circle if coefficient of x\(^{2}\) (i.e., a) = coefficient of y\(^{2}\) (i.e., b) and coefficient of xy (i.e., h) = 0.
Note: On comparing the general equation x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 of a circle with the general equation of second degree ax\(^{2}\) + 2hxy + by\(^{2}\) + 2gx + 2fy + C = 0 we find that it represents a circle if a = b i.e., coefficient of x\(^{2}\) = coefficient of y\(^{2}\) and h = 0 i.e., coefficient of xy.
The equation ax\(^{2}\) + ay\(^{2}\) + 2gx + 2fy + c = 0, a ≠ 0 also represents a circle.
This equation can be written as
x\(^{2}\) + y\(^{2}\) + 2\(\frac{g}{a}\)x + 2\(\frac{f}{a}\)y + \(\frac{c}{a}\) = 0
The coordinates of the centre are (\(\frac{g}{a}\), \(\frac{f}{a}\)) and radius \(\mathrm{\frac{1}{a}\sqrt{g^{2} + f^{2}  ca}}\).
Special features of the general equation ax\(^{2}\) + 2hxy + by\(^{2}\) + 2gx + 2fy + C = 0 of the circle are:
(i) It is a quadratic equation in both x and y.
(ii) Coefficient of x\(^{2}\) = Coefficient of y\(^{2}\). In solving problems it is advisable to keep the coefficient of x\(^{2}\) and y\(^{2}\) unity.
(iii) There is no term containing xy i.e., the coefficient of xy is zero.
(iv) It contains three arbitrary constants viz. g, f and c.
`11 and 12 Grade Math
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