Choice C is correct. In the $xy$ -plane, the graph of the line $y = c$ is a horizontal line that crosses the $y$-axis at $y = c$ and the graph of the quadratic equation $y = -x^2 + 9x - 100$ is a parabola. A parabola can intersect a horizontal line at exactly one point only at its vertex. Therefore, the value of $c$ should be equal to the $y$-coordinate of the vertex of the graph of the given equation. For a quadratic equation in vertex form, $y = a(x - h)^2 + k$, the vertex of its graph in the $xy$-plane is $\left(h, k\right)$. The given quadratic equation, $y = -x^2 + 9x - 100$, can be rewritten as $y = -( x^2 - 2(\dfrac{9}{2})x + (\dfrac{9}{2})^2) + (\dfrac{9}{2})^2 - 100$ or $y = -( x - \dfrac{9}{2})^2 + (-\dfrac{319}{4})$

Thus, the value of $c$ is equal to $-\dfrac{319}{4}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.