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Question

Find the centre and radius of each of the following circles:
(i) (x − 1)2 + y2 = 4
(ii) (x + 5)2 + (y + 1)2 = 9
(iii) x2 + y2 − 4x + 6y = 5
(iv) x2 + y2 − x + 2y − 3 = 0.

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Solution

Let (h, k) be the centre of a circle with radius a.
Thus, its equation will be x-h2+y-k2=a2.

(i) Given:
(x − 1)2 + y2 = 4

Here, h = 1, k = 0 and a = 2

Thus, the centre is (1, 0) and the radius is 2.

(ii) Given:
(x + 5)2 + (y + 1)2 = 9

Here, h = −5, k = −1 and radius = 3

Thus, the centre is (−5, −1) and the radius is 3.

(iii) Given:
x2+y2-4x+6y=5

The given equation can be rewritten as follows:
x-22+y+32-4-9=5
x-22+y+32=18

Thus, the centre is (2, −3).
And, radius = 18=32

(iv) Given:
x2+y2-x+2y-3=0

The given equation can be rewritten as follows:
x-122+y+12-14-1-3=0
x-122+y+12=174

Thus, the centre is 12,-1 and and the radius is 172.

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