Parabolas and circles BFFs!!

Let the two points be :
$\displaystyle A(t^2_1,2t_1)$ and $\displaystyle B(t^2_2,2t_2)$

Then, the Midpoint of these 2 points will represent the center of the circle (as they both are the end of its diameter). Thus,
If $\displaystyle O$ represents the center of the circle,
$\displaystyle O(\frac{t^2_1+t^2_2}{2},t_1+t_2)$

Now,
the axis of the Parabola is the line $\displaystyle y=0$ , therefore, for the circle to touch this line, the perpendicular distance of the center of the circle from this line must be equal to the radii.

Hence,
$\displaystyle t_1+t_2 = r$

Also, note that the slope of the line joining $\displaystyle A$ and $\displaystyle B$ is equal to $\displaystyle \frac{2}{t_1+t_2}$

Hence,
Slope $\displaystyle = \frac{2}{r}$

Considering the same case with the figure below x-axis, we will get the slope as negative, but of the same magnitude.

Therefore,
Slope $\displaystyle = \frac{\pm 2}{r}$