I've been trying to find a solution to this problem but I'm not too sure how to go about solving it.
I need to find the unknown values of $A, B$ and $C$ in the parabola equation $y(x) = Ax^2 + Bx + C$ given that the parabola passes through point (0,0) and is tangent to the line $y1(x) = 0.1x$ which also passes through the point (0,0).
I also need to find an unknown point (x-coordinate, y-coordinate) on the parabola which is tangent to $y2(x) = -0.08x + 10$ given that $y2(x) = -0.08x + 10$ passes through (200,-6).
I would greatly appreciate any help with solving this problem. So far I've tried to use the $y1(x)$ line to determine the vale of $B$ and $C$ in the parabola:
As the parabola passes through (0,0) I tried,
$$y(0) = 0$$ so $$A(0)^2 + B(0) + C = 0\\ C = 0$$
As the line $y1(x) = 0.1x$ is tangent to the parabola at (0,0) I tried,
$$ y'(0) = 0.1$$ so
$$ 2A(0) + B = 0.1\\ B = 0.1$$
However when trying to find a point on the parabola where the parabola is tangent to $y2(x)$ it seems to me that the value for B would be different, so this has me really confused on how to solve this problem.
Once again, any help with this would be very very appreciated :)