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.

● The Circle

• Definition of Circle
• Equation of a Circle
• General Form of the Equation of a Circle
• General Equation of Second Degree Represents a Circle
• Centre of the Circle Coincides with the Origin
• Circle Passes through the Origin
• Circle Touches x-axis
• Circle Touches y-axis
• Circle Touches both x-axis and y-axis
• Centre of the Circle on x-axis
• Centre of the Circle on y-axis
• Circle Passes through the Origin and Centre Lies on x-axis
• Circle Passes through the Origin and Centre Lies on y-axis
• Equation of a Circle when Line Segment Joining Two Given Points is a Diameter
• Equations of Concentric Circles
  
• Circle Passing Through Three Given Points
• Circle Through the Intersection of Two Circles
• Equation of the Common Chord of Two Circles
• Position of a Point with Respect to a Circle
• Intercepts on the Axes made by a Circle
• Circle Formulae
• Problems on Circle

11 and 12 Grade Math 

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