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Circle Formulae

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 = a2.

(ii) The general form of the equation of a circle is x2 + y2 + 2gx + 2fy + c = 0, where the co-ordinates of the centre are (-g, -f) and radius = g2+f2c units.

(iii) The equation of a circle with centre at the origin O and radius equals to ‘a’ is x2 + y2 = a2

(iv) The parametric form of the equation of the circle x2 + y2 = r2 is x = r cos θ, y = r sin θ.

(iv) The general second degree equation in x and y (ax2 + 2hxy + by2 + 2gx + 2fy + c = 0) represents a circle if coefficient of x2 (i.e., a) = coefficient of y2 (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 (x1, y1) and (x2, y2) as diameter is (x - x1)(x - x2) + (y - y1)(y - y2) = 0

(vi) A point (x1, y1) lies outside, on or inside a circle S = x2 + y2 + 2gx + 2fy + c = 0 according as S1 > = or <0, where S1 = x12 + y12 + 2gx1 + 2fy1 + c.

(vii) The equation of the common chord of the intersecting  circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0 is 2(g1 - g2) x + 2(f1 - f2) y + c1 - c2 = 0.

(viii) The equation of any circle through the points of intersection of the circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0 is x2 + y2 + 2g1 x + 2f1y + c1 + k (x2 + y2 + 2g2x + 2f2y + c2) = 0 (k ≠ -1).

(ix) The equation of a circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is  x2 + y2 + 2gx + 2fy + c' = 0.

(x) The lengths of intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 with X and Y axes are 2g2c and 2f2c respectively.

 The Circle




11 and 12 Grade Math 

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