Given the $y$-intercept and gradient of an infinite straight line on the Cartesian plane, how can I find both points of intersection (2D vector) if the line does pass through the curve $x^2 + y^2 = 16$ in terms of $m$ and $c$. In addition, I would like to find the point equidistant from both intersections.
Thanks, Ben
Hint:
Points of intersection occur when both equations are satisfied: meaning, all $(x,y)$ such that both equations \begin{align} x^2 + y^2 &= 16\\ y &= mx + c \end{align} hold. You can substitute the second equation into the first, giving a quadratic in $x$. Depending on what $m$ and $c$ are, you can have 0 to 2 real solutions to this quadratic: either the line doesn't touch the circle at all (0 solutions), is tangent to it (1 solution) or passes through twice (2 solutions).
If there are two solutions $(x_1,y_1)$ and $(x_2,y_2)$ the point equidistant is just averaging both coordinates: $$ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). $$