A line segment $A M = a$ moves in the $X O Y$ plane such that $A M$ is parallel to the $X$ -axis. If $A$ moves along the circle $x^{2} + y^{2} = a^{2}$ , then the locus of $M$ is

x^{2} + y^{2} = 4 a^{2}

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x^{2} + y^{2} = 2 a x

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x^{2} + y^{2} = 2 a y

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x^{2} + y^{2} = 2 a x + 2 a y

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Solution

The correct option is B $x^{2} + y^{2} = 2 a x$
Given the equation of the circle: $S_{1} : x^{2} + y^{2} = a^{2}$
If $A$ moves along this $S_{1}$ and $A M$ always remain parallel to $X -$ axis, then visualizing the figure, we can say that locus of $M$ will be the circle with centre $(a, 0)$ and radius equal to length of $A M$ that is $a$
Therefore, locus of $A M$ is $(x - a)^{2} + y^{2} = a^{2}$
$\Rightarrow x^{2} + y^{2} = 2 a x$

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List I	List II
A) An equation of the circle touching the x- axis is	1) $x^{2} + y^{2} + 2 a x + 2 a y + a^{2} = 0$
B) An equation of the circle touching the y- axis is	2) $x^{2} + y^{2} - 2 a x + 2 b y + b^{2} = 0 y$
C) An equation of the circle touching both the axes is	3) $x^{2} + y^{2} + 2 a x - 2 a y + a^{2} = 0$
D) An equation of the circle passing through (a, b) is	4) $x^{2} + y^{2} + a x - b y - 2 a^{2} = 0$

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