Amaths - Locus problem

Let M (x1,y1) N (x1,y2)

So the mid-point of MN P (x1, (y1+y2)/2 )

Now
y1=4x1^2+3
y2=(24-4x1)/3

Now (y1+y2)/2
=(1/2)[4x1^2+3+(24-4x1)/3]
=(1/6)[12x1^2+9+24-4x1]
=(1/6)[12x1^2-4x1+33]

So now substitute x=x1 and y=(y1+y2)/2, we get the equation of the locus of the mid-point of MN when the line moves, which is

y=(1/6)[12x^2-4x+33]

12x^2-4x-6y+33=0