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Question
Answer the following :
Find the equation of the circle concentric with x2 + y2 – 4x + 6y = 1 and having radius 4 units
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Solution
Given equation of circle is
x2 + y2 – 4x + 6y = 1
i.e., x2 + y2 – 4x + 6y – 1 = 0
Comparing this equation with
x2 + y2 + 2gx + 2fy + c = 0, we get
2g = –4, 2f = 6
∴ g = –2, f = 3
∴ Centre of the circle = (–g, –f) = (2, –3)
Given circle is concentric with the required circle.
∴ They have same centre.
∴ Centre of the required circle = (2, –3)
The equation of a circle with centre at (h, k) and radius r is
(x – h)2 + (y – k)2 = r2
Here, h = 2, k = –3 and r = 4
∴ the required equation of the circle is
(x – 2)2 + [y – (–3)]2 = 42
∴ (x – 2)2 + (y + 3)2 = 16
∴ x2 – 4x + 4 + y2 + 6y + 9 – 16 = 0
∴ x2 + y2 – 4x + 6y – 3 = 0.
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