Proving that a point P is on a circle and a line

Question

1. Given a circle $C$ with center $A$ and radius $r$.
2. Given a line $D$ with a vector ${u}$ passing through point $P_0$.
3. Knowing that $P$ is on $D$ only if $P = P_0 + t u$
4. Knowing that $P$ is on $C$ if $\lVert{P - A}^2\rVert = r^2$

Prove that point $P$ is on $C$ and $D$ if there exists a real number $t$ where \begin{equation} \left[\lVert u \rVert ^2\right] t^2 + \left[2(P_0 - A) \cdot u\right]t + \left[\lVert{P_0 - A}^2\rVert - r^2\right] = 0. \end{equation}

What properties should I be using in order to solve this?

Answer 1

That equation is what we get when we expand out $\|P_0+tu-A\|^2=r^2$: $$\|P_0+tu-A\|^2=(tu+P_0-A)\cdot (tu+P_0-A) = t^2\|u\|^2+2tu\cdot (P_0-A)+\|P_0-A\|^2$$ Set that equal to $r^2$, rearrange a bit, and there it is.