1. **State the problem:** We have a line with equation $y = mx + c$ and a curve with equation $xy = 16$. The line is tangent to the curve, and we need to express $m$ in terms of $c$. 2. **Rewrite the curve equation:** From $xy = 16$, express $y$ as a function of $x$: $$y = \frac{16}{x}$$ 3. **Condition for tangency:** The line $y = mx + c$ touches the curve $y = \frac{16}{x}$ at exactly one point. At the point of tangency, the two equations are equal: $$mx + c = \frac{16}{x}$$ Multiply both sides by $x$ to clear the denominator: $$x(mx + c) = 16$$ $$m x^2 + c x = 16$$ 4. **Rewrite as a quadratic in $x$:** $$m x^2 + c x - 16 = 0$$ 5. **Tangency means one solution:** For the quadratic to have exactly one solution, its discriminant must be zero: $$\Delta = b^2 - 4ac = 0$$ Here, $a = m$, $b = c$, and $c = -16$ (constant term). So: $$c^2 - 4 m (-16) = 0$$ $$c^2 + 64 m = 0$$ 6. **Solve for $m$ in terms of $c$:** $$64 m = -c^2$$ $$m = -\frac{c^2}{64}$$ **Final answer:** $$\boxed{m = -\frac{c^2}{64}}$$