Find a unit tangent vector to a curve that is an intersection of two surfaces. The intersection of the two surfaces given by the Cartesian equations $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ passing through the point $P=(\sqrt{7},3,4)$. These equations may be solved for $x$ and $y$ in terms of $z$ to give a parametric representation of $C$ with $z$ as a parameter.
(a) Find a unit tangent vector $T$ to $C$ at the point $P$ without using an explicit knowledge of the parametric representation.
(b) Check the result in part (a) by determining a parametric representation of $C$ with $z$ as a parameter.
For (b), I solved the two equations for the surfaces given to get $y^2=25-z^2$ and $x^2=2z^2-25$. Since we're looking for the curve that contains $P$, $x$, $y$ should be positive so we get $y=\sqrt{25-z^2}$ and $x=\sqrt{2z^2-25}$. So from this I get the parametric representation $(\sqrt{2z^2-25}, \sqrt{25-z^2},z)$ for the curve $C$. 
Is this the correct way of finding the parametrization? Moreover, I do not know how to find the unit targent vector $T$ to $C$ at $P$, without getting a parametrization. How can I find this? The answer to $T$ is $\frac{1}{\sqrt{751}}(24,-4\sqrt{7},3\sqrt{7})$. 
I would greatly appreciate any solutions, hints or suggestions.

For $(b)$, by
$$
\begin{cases}
2x^2+3y^2-z^2=25 \\
x^2+y^2=z^2 \\
\end{cases}
$$
and working on the first:
$$\underbrace{2x^2+2y^2}_{2 z^2}+y^2-z^2=25\ \Rightarrow \ z^2+y^2=25$$
Then the curve is a circumference in the y-z plane and its parametric representation is:
$$
\begin{cases}
x=5 \cos t \\
y=5 \sin t \\
\end{cases}
$$
for $t\in[0,2\pi]$.
