How do I solve this quadratic equation in terms of t?

Question

I'm having trouble with this equation. I thought it would be easy to solve using the quadratic formula, but I have no idea how to start. This is not homework, but I would like to know how to solve this in terms of $t$. Both sides are pretty much the same since it has to do with two points $(a,b)$ and $(c,d)$ which are equidistant to the point $(t,\sqrt{1-t^2})$, but this information is not relevant to the problem. How do I isolate $t$ in this situation:

$a^2-2b\cdot \sqrt{1-t^2}+b^2+2ct=c^2-2d\cdot \sqrt{1-t^2}+d^2+2at$

Answer 1

HINT: your equation can be written in the form $$a^2+b^2-c^2-d^2+(2c-2a)t=\sqrt{1-t^2}(2b-2c)$$ we denote by $$A=a^2+b^2-c^2-d^2$$ $$B=(2c-2a)$$ $$C=2b-2d$$ then we have $$A+Bt=\sqrt{1-t^2}C$$ after squaring we get $$t^2(B^2+C^2)-2ABt+A^2-C^2=0$$ can you solve this?

Answer 2

The solution of your problem are the points of the unit circle $u$ that are equidistant to the points $P (a,b)$ and $Q (c,d)$. These can be found by intersecting $u$ with the line segment bisector of the segment $\overline {PQ}$.