Question

Quadratic equations of concentric circles and roots, find the center of the circle

Answer 1

Two circles or more than that are said to be concentric if they have the same centre but different radii.

Let, x2 + y2 + 2gx + 2fy + c = 0 be a given circle having centre at (- g, - f) and radius =

√

g2+f2−c

.

Therefore, the equation of a circle concentric with the given circle x2 + y2 + 2gx + 2fy + c = 0 is

x2 + y2 + 2gx + 2fy + c' = 0

Both the circle have the same centre (- g, - f) but their radii are not equal (since, c ≠ c')

Similarly, the equation of a circle with centre at (h, k) and radius equal to r, is (x - h)2 + (y - k)2 = r2.

Therefore, the equation of a circle concentric with the circle (x - h)2 + (y - k)2 = r2 is (x - h)2 + (y - k)2 = r12, (r1 ≠ r)

Assigning different values to r1 we shall have a family of circles each of which is concentric with the circle (x - h)2 + (y - k)2 = r2.

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