Question:
If $(-3, 2)$ lies on the circle $x^2 + y^2 + 2gx + 2fy + c = 0$, which is concentric with the circle $x^2 + y^2 + 6x + 8y - 5 = 0,$ then c is equal to
If $(-3, 2)$ lies on the circle $x^2 + y^2 + 2gx + 2fy + c = 0$, which is concentric with the circle $x^2 + y^2 + 6x + 8y - 5 = 0,$ then c is equal to
Updated On: Apr 28, 2024
- 11
- -11
- 24
- 100
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The Correct Option is B
Solution and Explanation
Equation of family of concentric circles to the circle $x^2 + y^2 + 6x + 8y - 5 = 0$ is
$x^2 + y^2 + 6x + 8y + \lambda = 0$
which is similar to
$x^2 + y^2 + 2gx + 2fy + c = 0$
Thus, the point $(-3, 2)$ lies on the cirlce
$x^2 + y^2 + 6x + 8y + c = 0$
$\therefore \, (-3)^2 + (2)^2 + 6(-3) + 8 (2) + c = 0$
$\Rightarrow \, 9 + 4 - 18 + 16 + c = 0 $
$ \Rightarrow c = -11$
$x^2 + y^2 + 6x + 8y + \lambda = 0$
which is similar to
$x^2 + y^2 + 2gx + 2fy + c = 0$
Thus, the point $(-3, 2)$ lies on the cirlce
$x^2 + y^2 + 6x + 8y + c = 0$
$\therefore \, (-3)^2 + (2)^2 + 6(-3) + 8 (2) + c = 0$
$\Rightarrow \, 9 + 4 - 18 + 16 + c = 0 $
$ \Rightarrow c = -11$
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