There is a circle of radius 187.5m and centre $(25.23,188.6)$. It takes the form $(x-25.23)^2 +(y-188.6)^2=187.5^2$
There is another circle whose centre is unknown and has a radius of 1750m. This circle intersects with the y axis at $(0,2.05)$ and a point on the original circle. The point on the original circle is the point where the tangent line from point $(0,2.05)$ touches the larger circle. I want to obtain the centre of this circle.
I've created this graphically in CAD software, and the resultant centre should be close to $(87.5028,1749.861)$. I've tried doing this mathematically but i fear I haven't dealt with quadratics in a long time. Sometimes I get close to the point provided by CAD but when verifying the circle equation by plugging the point $(0,2.05)$ back in, it doesn't converge.
I've calculated the tangent point in this case to be $(16.752,1.29$) which I believe is correct. However my problem lies when trying to get the centre point of the larger circle. If both points are on the circle then the following is true:
$(0-h)^2 + (2.05-k)^2 = 1750^2$ and $(16.752-h)^2 + (1.29-k)^2 = 1750^2$
I solve for h in terms of k in one equation. Reinput h back into the other equation and solve for k. Once I solve for the values of k, I then solve for h. But I've used online calculators and still can't seem to reconcile the numbers.
Could someone possibly explain if my work is flawed in some way? Or possibly identify that it is possible to get similar numbers to the graphical solution? I would provide the graphic but the scales of the circles are so large it's difficult to see the area I'm concentrating on. Thanks