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

Options

A. `y=3/4`
B. `y=-3/4`
C. `y=x^(2)`
D. `y=-x^(2)`

Answer

A

Explanation

Let `F` be the fixed focus and `M` be the moving focus and `T` be the varying points of mutual tangency. The tangent line at `T` makes equal angle with `FT` and with as vertical line. This and congruence of the two parabola imply that `MT` is vertical and `FT=MT` and must line one directrix of `y=x^(2)-x+1`
So, `y=3/4`