Question

Consider two parabola y = x² – x + 1 and the parabola y= - x² + x + 1/2 which is fixed, and parabola y = x²– x + 1 rolls without slipping around the fixed parabola, then the locus of the focus of the moving parabola is?

Answer 1

for the parabola, y=a(x−h)²+k, then the vertex is at (h,k) and the focus is (h,k+1/4a)

consider y = x²– x + 1 is the rolling parabola
y = (1)(x-1/2)² + 3/4  where a=1
and y= - x² + x + 1/2 is the fixed parabola
then (x₁, - x₁² + x₁ + 1/2) is the vertex of rolling parabola whose a=1
so the resulting parabola with vertex(x₁, - x₁² + x₁ + 1/2) and a=1
y=(x−x₁)²+(- x₁² + x₁ + 1/2)
focus (h,k+1/4a) = (x₁, (- x₁² + x₁ + 1/2)+1/4)
y=(- x₁² + x₁ + 1/2)+1/4
so the locus is y=(- x² + x + 3/4)



 i hope i am right (my gut says otherwise- i think i should take a tangent then find the vertex to the rolling parabola)