I'd like to find a way of determining if the distance from the origin of a parametric parabolic path falls below a certain value within a given range of the parameter. The parabola is expressed as:

$$x = a_xt^2+b_xt+c_x$$ $$y = a_yt^2+b_yt+c_y$$

So I'm looking for $x^2+y^2=d^2$ for $t_1 \leq t \leq t_2$.

Obviously I can plug in the first $2$ equations into the third, solve the quartic and test if any of the four solutions for $t$ lies within the range, but I would like to see if there is a faster test for this. My idea is to formulate it as a constrained optimization problem with constraints $x^2+y^2-d^2=0$, $-t \leq t_1$ and $t \leq t_2$ but I'm not sure how to proceed. I don't need the to know the exact minimum point, just whether the distance falls below $d$ with the range.

Am I on the wrong path?

Thanks.