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Let P be a variable point on the parabol...

Let P be a variable point on the parabola `y = 4x ^(2) +1.` Then, the locus of the mid- point of the point P and the foot of the perpendicular drawn from the point P to the line y =x is:

A

`( 3x - y) ^(2) + ( 3 - 3y ) + 2 =0`

B

`2 ( 3x -y) ^(2) + ( x - 3y) + 2=0`

C

`2 (x - 3y) ^(2) + ( 3x - y) + 2 = 0`

D

`( 3x - y) ^(2) + 2 (x - 3y) + 2 =0`

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