A moving point remains equidistant from the point (a,0)and the line y=x. Find the equation of its locus.

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We need to determine the locus of a point that is equidistant from the line y = x and the point (a , 0)

Now the relation for determining the distance d of a point (x1, y1) from the line ax+by +c = 0, is:

d = |ax1+by1+c|/ sqrt (a^2+b^2)

and the relation for the distance between two points (x1, y1) and (x2, y2) is sqrt [ (x2 - x1)^2 + (y2 - y1)^2]

Let the required point be (X, Y). Substituting the values we have:

| X - Y | / sqrt ( 1^2 + 1^2) = sqrt [( X - a)^2 + Y^2]

=> |X - Y| / sqrt 2 = sqrt [( X - a)^2 + Y^2]

=> |X - Y| = sqrt [ 2* (X - a)^2 + 2*Y^2]

Therefore the required locus is:

x - y = sqrt [2(x - a)^2 + 2y^2]

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