The condition if the equation of a circle concentric with the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is $x^{2} + y^{2} + 2 g x + 2 f y + k = 0$ , where $k$ is a constant.

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Solution

Given equations:
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0 \to (I)$
$∴$ Centre $\to$ $(- g, - f)$
and radius $= \sqrt{g^{2} + f^{2} - c}$
and $x^{2} + y^{2} + 2 g x + 2 f y + k = 0 \to (I I)$
Again, centre $\to$ $(- g, - f)$
and radius $= \sqrt{g^{2} + f^{2} - k}$
Same centre and different radii,
hence concentric circles.
$∴$ Given statement is true.
$(i f k = c) .$

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The condition if the equation of a circle concentric with the circle x2+y2+2gx+2fy+c=0 is x2+y2+2gx+2fy+k=0, where k is a constant.

Given equations:x2+y2+2gx+2fy+c=0→(I)∴ Centre → (−g,−f) and radius =√g2+f2−cand x2+y2+2gx+2fy+k=0→(II) Again, centre → (−g,−f)and radius =√g2+f2−kSame centre and different radii,hence concentric circles.∴ Given statement is true.(ifk=c).

The condition if the equation of a circle concentric with the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is $x^{2} + y^{2} + 2 g x + 2 f y + k = 0$ , where $k$ is a constant.