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**HallsofIvy** What do you mean by "the point on the section of OP nearer the origin"? Nearer the origin than what? What do you mean by "section of OP"? OP normally means "the line segment from O to P" and that includes O itself.

What, exactly, did you do to find Q= (2x/3, 2y/3)? For that matter, what does it mean? Is the "y" in that given by y= 2x^2+ 3? If so, then (2x/3, 2y/3)= (2x/3, 4x^2/3+ 2). Now, let u= 2x/3 so that x= 3u/2. Putting that into y= 4x^2/3+ 2 gives y= 4(9u^2/4)+ 2= 9u^2+ 2 or (u, 9u^2+2). That gives y= 9x^2+ 2 as the locus.