Question

Answer the following :

Find the equation of the circle concentric with x² + y² – 4x + 6y = 1 and having radius 4 units

Answer 1

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.