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Equation of family of circles

1.  The equation of circle with centre A  (h, k) and radius AP = r :  (x - h)2 + (y - k)2 = r2

    

2.  The equation of circle with centre at origin 0(0, 0) and radius = r:

x2 + y2 = r2

 

3.  The equation of circle with centre (-g, -f) and radius = r :

x2 + y2 +2gx +2fy +c = 0, where c = g2 + f2 - r2

 

This is also known as general form of equation of circle. For this circle:

(i)         Centre is ( - g , -f)   

 

(ii)        Radius = r =

 

Different types of Circles

 

(i)  Centered at X-axis:

=>   Y-coordinate of centre  0, i.e., f = 0

=>   centre is  (-g , 0)

=>   x2 + y2 + 2gx + c = 0

 

 

(ii)  Centered at Y-axis:

=>   X-coordinate of centre  0, i.e., g = 0

=>   centre is.  (0,-f)

=>   x2 + y2 + 2fy + c = 0

 

(iii)  Centered at origin:

=>   center is  (-g, -f)  (0, 0)

=>   x2 + y2 + c = 0 is the equation of circle. (c = -r2)

(iv) Circle passing through origin:

i.e., (0, 0) must lie on the circle or (0, 0) satisfies the equation of circle.

=>   c = 0

=>   x2 + y2 + 2gx + 2fy = 0

 

 

Note:      

A circle x2 + y2 - ax - by =0 passes through origin and cuts X-axis at (a, 0) and Y-axis at (0, b).

(v) Circle centered at X-axis and touching Y-axis at origin:

                x2 + y2 + 2gx = 0 is the circle.

 

                =>   (0, 0) and (-2g, 0) are end points of diameter.

                               
                  

 

(vi) Circle centered at Y-axis and touching X-axis at origin:

                x2 + y2 + 2fy = 0 is the circle.

 

=>   (0, 0) and (0, -2f) are end points of diameter.

                                     

                                  

(vii) Circle touching both the axes:

The centre of circle of radius r touching both axes OX and OY is:

(± r, ± r)

(x  r)2 + (y  r)2 = r2

 

(viii)  Equation of circle in diametric form:

(x - x1) (x - x2) + (y - y1) (y - y2) = 0 is the equation of circle,

 

where (x1, y1) and (x2, y2) are the end points of any diameter. 

(ix) Parametric form of circle:

Consider the circle (x - h)2 + (y - k)2 = r2 centered at A   (h, k) and of radius r.

                                     

 

Let P (x, y) be any point on this circle. Say AP has a slope of tan (). Thus coordinates of P can be expressed as:

 (h + projection of AP on x axis)

 (k + projection of AP on Y axis)

 

These two equations represent the coordinates of any point on the circle in terms of the parameter θ.

 

Thus for a circle centered at origin, the parametric form of equations are:





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