鉜4 (Yellow). Find the tangent line to the hyperbola $\frac{x^{2}}{64}-\frac{y^{2}}{16}=1$ passing through the point $P(16,8)$.

 untranslated text remains the same as it is a formula or point notation.
Substitute $(16,8)$ into the tangent line equation $\mathbf{y}=\mathrm{k} \mathbf{x} \pm \sqrt{ } 64 \overline{\mathrm{k}^{2}-16}$ $k_{1}=\frac{5}{6}, k_{2}=\frac{1}{2}$, hence the tangents are $y=\frac{5}{6} x-\frac{16}{3}\left(k>\frac{1}{2}\right.$ when taking “ - ” $)$ and $y=\frac{1}{2} x$.

Analysis: Due to neglecting that the slope $\mathbf{k}$ of the hyperbola should satisfy $|k|>\frac{b}{a}=\frac{1}{2}$, the asymptote $\mathrm{y}=\frac{1}{2} x$ was mistakenly considered as a “tangent” in the problem.