(A) Let the point be $P(x, y)$.The distance between $P(x, y)$ and $(-g, -f)$ is given as $a$.Using the distance formula: $\sqrt{(x - (-g))^2 + (y - (-

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$x^2 + y^2 + 2gx + 2fy + k = 0$

(A) Let the point be $P(x, y)$.The distance between $P(x, y)$ and $(-g, -f)$ is given as $a$.Using the distance formula: $\sqrt{(x - (-g))^2 + (y - (-f))^2} = a$Squaring both sides: $(x + g)^2 + (y + f)^2 = a^2$Expanding the squares: $x^2 + 2gx + g^2 + y^2 + 2fy + f^2 = a^2$Rearranging the terms: $x^2 + y^2 + 2gx + 2fy + g^2 + f^2 - a^2 = 0$Given that $k = g^2 + f^2 - a^2$,we substitute $k$ into the equation:$x^2 + y^2 + 2gx + 2fy + k = 0$.

Class 11 Mathematics 10-1.Circle and System of Circles Locus Related Problem

Medium

The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is:

A
$x^2 + y^2 + 2gx + 2fy + k = 0$
B
$x^2 - y^2 + 2gx + 2fy + k = 0$
C
$x^2 + y^2 + 2xy + 2gx + 2fy + k = 0$
D
None of these

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10-1.Circle and System of Circles

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Locus Related Problem

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The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is:

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