The number of points on an elliptic curve with square x-coordinates

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Let K be a finite field. We know that a half of elements of K* is a square. So it is natural to ask how many of them appear as x-coordinate of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite fields and show that a half of x-coordinate on an elliptic curve is a square. This result generalizes my old paper posted 30 Dec 2009.