Let P have coordinates . Then the center of OP has coordinates . The slope of the chord is , so the slope of the perpendicular is . Thus, the equation of the perpendicular bisector is , and . From here, . Therefore, and not as I thought initially.
I found the following theorem in this PDF document.
The axis is the vertical one. Indeed, the length of the segment in question is , the focus of the parabola is and the directrix is .The perpendicular bisector of a parabola chord and the perpendicular to the axis through the center of the chord cut off a segment on the axis of length equal to the distance of the focus to the directrix.