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)$$^{2}$$ = 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.