Find coordinates of point that intersects circle

Question

I got a circle of 900 radius, knowing its center coordinates A(x1, y1) and got another point with also known coordinates B(x2, y2). I draw a line between point A and B. It intersects the circle in a point C(x,y). How can I find the coordinates of that point?

Answer 1

Let $\Delta x = x_2 - x_1$ and $\Delta y = y_2 - y_1$. Then by constructing similar right triangles and applying Pythagoras, it follows that: $$ (x, y) = \left(x_1 + \frac{900 \Delta x}{\sqrt{(\Delta x)^2 + (\Delta y)^2}},\quad y_1 + \frac{900 \Delta y}{\sqrt{(\Delta x)^2 + (\Delta y)^2}} \right) $$

Answer 2

Find the equation of the line $AB$, the equation of the circle, form a system and solve it. You'll get two solutions, say $P$ and $Q$ (because the line cuts the circle in two points: you want the one between $A$ and $B$).

Find the distantes $PB$ and $QB$. The shortest corresponds to the true solution.

Answer 3

Hints:

1. Think about the distance from the center of the circle to that point.
2. Think about the slope of the line, and how to find it.
3. Convert that slope into an angle.

If you have an angle and a length, you should be able to draw a right triangle, and find the lengths of its other two sides. You can use these sides for your coordinates.