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\( A B C D \) is a square away from origin of side length ' \( a \) '. Its side \( A B \) slides between \( x \) and \( y \)-axes in first quadrant with \( A \) on \( x \)-axis and \( B \) on \( y \)-axis. The locus of the foot of perpendicular dropped from the point
\( \mathrm{P} \) \( E \) on the diagonal \( A C \) (where \( E \) is the midpoint of the side \( A D \) ), is
(A) \( (y-x)^{2}+(x-3 y)^{2}=a^{2} \)
(B) \( (y-x)^{2}+(x-3 y)^{2}=\frac{a^{2}}{2} \)
(C) \( (y-x)^{2}+(x-3 y)^{2}=\frac{a^{2}}{4} \)
(D) None of these
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