Circle formulae will help us to solve different types of problems on circle in co-ordinate geometry.
(i) The equation of a circle with centre at (h, k) and radius equals to ‘a’ units is (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\).
(ii) The general form of the equation of a circle is x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0, where the co-ordinates of the centre are (-g, -f) and radius = \(\mathrm{\sqrt{g^{2} + f^{2} - c}}\) units.
(iii) The equation of a circle with centre at the origin O and radius equals to ‘a’ is x\(^{2}\) + y\(^{2}\) = a\(^{2}\)
(iv) The parametric form of the equation of the circle x\(^{2}\) + y\(^{2}\) = r\(^{2}\) is x = r cos θ, y = r sin θ.
(iv) The general second degree
equation in x and y (ax\(^{2}\) + 2hxy + by\(^{2}\) + 2gx + 2fy + c = 0) 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.
(v) The equation of the circle drawn on the straight line joining two given points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) as diameter is (x - x\(_{1}\))(x - x\(_{2}\)) + (y - y\(_{1}\))(y - y\(_{2}\)) = 0
(vi) A point (x\(_{1}\), y\(_{1}\)) lies outside, on or inside a circle S = x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 according as S\(_{1}\) > = or <0, where S\(_{1}\) = x\(_{1}\)\(^{2}\) + y\(_{1}\)\(^{2}\) + 2gx\(_{1}\) + 2fy\(_{1}\) + c.
(vii) The equation of the common chord of the intersecting circles x\(^{2}\) + y\(^{2}\) + 2g\(_{1}\)x + 2f\(_{1}\)y + c\(_{1}\) = 0 and x\(^{2}\) + y\(^{2}\) + 2g\(_{2}\)x + 2f\(_{2}\)y + c\(_{2}\) = 0 is 2(g\(_{1}\) - g\(_{2}\)) x + 2(f\(_{1}\) - f\(_{2}\)) y + c\(_{1}\) - c\(_{2}\) = 0.
(viii) The equation of any circle through the points of intersection of the circles x\(^{2}\) + y\(^{2}\) + 2g\(_{1}\)x + 2f\(_{1}\)y + c\(_{1}\) = 0 and x\(^{2}\) + y\(^{2}\) + 2g\(_{2}\)x + 2f\(_{2}\)y + c\(_{2}\) = 0 is x\(^{2}\) + y\(^{2}\) + 2g\(_{1}\) x + 2f\(_{1}\)y + c\(_{1}\) + k (x\(^{2}\) + y\(^{2}\) + 2g\(_{2}\)x + 2f\(_{2}\)y + c\(_{2}\)) = 0 (k ≠ -1).
(ix) The equation of a circle concentric with the circle x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 is x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c' = 0.
(x) The lengths of intercepts made by the circle x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 with X and Y axes are 2\(\mathrm{\sqrt{g^{2} - c}}\) and 2\(\mathrm{\sqrt{f^{2} - c}}\) respectively.
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