# Distance between a point inside a circle and circumference on a line along a point outside of circle

Point $$A$$ is located outside of a circle centered at $$C$$ and with radius $$r$$. Point $$B$$ is given point inside the circle. How to calculate $$d$$, the length of line segment between $$B$$ and circumference on the line $$\overline{AB}$$. Would there be a solution regardless of where $$B$$ is inside the circle?

Update: given to the problem is coordinates $$A$$, $$B$$ and $$r$$. I will need to extend the same problem to 3D geometry with a sphere and 3D coordinates. I would appreciate any help.

• What do you mean by "calculate", what is given in the problem? Why couldn't you just draw AB, take the intersection with the circle and calculate the distance? Commented Jan 29, 2021 at 21:08
• Are you looking for a formula in terms of the coordinates of A, B, C, and the radius r?
– user694818
Commented Jan 29, 2021 at 21:09
• @MatthewDaly Yes I am trying to find a closed form solution (formula) in terms of given coordinates $A(x,y)$, $B(x,y)$ and $r$. I next need to extend the same solution to sphere and 3D coordinates.
– fhm
Commented Jan 29, 2021 at 21:36

A method for you to use.

Yes there is always a solution. To find it you can do the following, where the centre of the circle is taken as the origin.

You can find the equation of the line $$AB$$ in the form $$y=mx+c$$.

Then the point $$P(x,y)$$ where the line crosses the circle satisfies the equation $$x^2+(mx+c)^2=r^2.$$

Solve this equation for $$x$$ and then find $$y$$ from $$y=mx+c$$. There will be two points, choose the one on the same side of the circle as $$A$$.

Finally calculate the distance $$BP$$.

• Thanks, would it be exactly the same if the 2D line and circle equations get converted to 3D coordinate ? I mean does the logic hold on a line go through a sphere ?
– fhm
Commented Jan 31, 2021 at 18:18
• The logic is the same. If you use the form of the equation of the line which has a parameter, t say, e.g. $x=4t+5,y=6t-8,z=9t$, then the $r^2$ equation can be solved for $t$ etc.
– user502266
Commented Jan 31, 2021 at 18:22