Details about Equation of C0-Axial system of circles?

Answer 1

Circles concentric about the centre (a,b) has a general equation
(x-a)^2 + (y-b)^2 = k*r^2

k can assume any positive value. This gives the equations for the family of concentric circles with centre at (a,b).

By the way concentric circles means circles with same centre but different radii.

Answer 2

The equation of a circle is (x-h)² + (y-k)² = r² where (h,k) is the coordinate of the center of the circle and r is the radius of the circle. It's pretty easy to see that, for the circles to be coaxial (have the same center), the h and k have to stay the same. But, as the r varies, the radius of the circle varies as well which leads to a family of coaxial circles with common center(s) at (h,k)


Doug

Answer 3

The equation of a circle centre (h,k) amd radius r is given by:
(x-h)² + (y-k)² = r²

So concentric circles are of the form
(x-h)² + (y-k)² = ar² wher a > 0

On expanding we see that x - 2hx + h² + y² - 2ky + k² - ar² = o

ie x² + y² - 2hx - 2ky + c = 0

ie the onlly variation in the family of coaxial circles is the value of the constant c (c = h² + k² - ar²; a >0)

Answer 4

equation of coaxial system of circles is
S+KS'=0
S+KT=0

where S =equation of 1 circle
S'=equation of 2nd circle
T=equation of radical axis of the two given circles
or
equation osf tangent to S
or
equation of a common chord

K=any scalar

Answer 5

(x-a)^2 +(y-b)^2 = r^2 provides a family of coaxicial circles normal to a line normal to the x-y plane passing through (a,b)