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Question
If (−3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which is concentric with the circle x2 + y2 + 6x + 8y − 5 = 0, then c =
Options
11
-11
24
none of these
Solution
−11
The centre of the circle x2 + y2 + 6x + 8y − 5 = 0 is (−3, −4).
The circle x2 + y2 + 2gx + 2fy + c = 0 is concentric with the circle x2 + y2 + 6x + 8y − 5 = 0.
Thus, the centre of x2 + y2 + 2gx + 2fy + c = 0 is (−3, −4).
\[\therefore g = 3, f = 4\]
Also, it is given that (−3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0.
\[\left( - 3 \right)^2 + 2^2 + 2\left( 3 \right)\left( - 3 \right) + 2\left( 4 \right)\left( 2 \right) + c = 0\]
\[\Rightarrow 9 + 4 - 18 + 16 + c = 0\]
\[ \Rightarrow c = - 11\]
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