Find Coordinates on circles circumference.

Question

i am having some trouble with some maths that i am dealing with.

To be more specific, i have 2 circles :

• first circle O with center (Xo,Yo)
• Radius R1
• i also have a point A(Xa,Ya) on that circles circumference

from the 2 circle J i know that:

• the center is the same as the firsts circle
• and the Radius R2 is R2=R1+k, where k is known

Now what i want is to find the points coordinates on the 2nd circle, where if we drew a line that starts from the center of the circle, passes the point A from the 1st circle and it stops on the 2nd circle. How can i find it out ? Anything would be helpfull, as i have been struggling with this for 3 days and found nothing.

Thanks in advance

Answer 1

You can write the equation of the line that goes through $A$ and $O$ as $$\frac{x-x_O}{x_A-x_O}=\frac{y-y_O}{y_A-y_O}$$ You know that $B$ is on this line, so $$\frac{x_B-x_O}{x_A-x_O}=\frac{y_B-y_O}{y_A-y_O}$$ So you have an equation with two unknowns, $x_B$ and $y_B$. The other equation is $$\frac{x_B-x_O}{x_A-x_O}=\frac{y_B-y_O}{y_A-y_O}=\frac{R_2}{R_1}$$ You can now solve the problem

Answer 2

Notice that the angle,α, –which is simple to calculate, α=arctan(Ya/Xa)– that separates the point A and the other point you're looking for, P(Px,Py), from either axis is the same for both points.

Now, if you choose the centre of the circles as the origin of coordinates, you'll see that the distance that separates P from the origin is R2=R1+k.

From this, you can construct a right triangle, with the origin and point P, whose hypotenuse is (R1+k) and small angle is α. Thus, the side opposite α will be Py, and the remaining one will be Px.

With a bit of trigonometry is easy to see that Px=(R1+k)*cos(α) and Py=(R1+k)*sin(α). In other words, you'll know the coordinates of P((R1+k)*cos(α), (R1+k)*sin(α))

See attached drawing for a more visual understanding and less wordiness.