How can I find the co-ordinate of where a line intersects a circle?

Question

I was looking to know if there was an equation that would allow me to calculate the co-ordinates of a point on the circumference of a circle where a line intersects it and the center. My diagram should display this better:

So how would I know the point of $B$ if I know the co-ordinates of $A$ and $C$ and also know the equation of the line? I also know the radius of the circle.

Answer 1

Hint: The coordinates of the point B which is the mid point of $A(x_A, y_A)$ & $C(x_C, y_C)$ are calculated as follows $$B\equiv\left(\frac{x_A+x_C}{2}, \frac{y_A+y_C}{2}\right)$$ Points $A$, $B$ & $C$ will lie on the line & will satisfy its equation.

Answer 2

If instead of bisection, you know the circle radius $r$ and the distance AC is $\ell$ then

$$ B \equiv \left(x_A + \frac{r}{\ell} (x_C-x_A), y_A + \frac{r}{\ell} (y_C-y_A) \right) $$

To get the bisection, use simply $r=\frac{\ell}{2}$.

Answer 3

simple analytic geometrical way:

$(x-A_x)^2 + (y-A_y)^2 = r^2$

$y = kx$

the desired point (x,y) satisfies both equations. Solve for x,y and you'll get two points (the line intersects at both ends), choose the upper one.