A line in the xy -plane has a slope of 1 9 and passes through the

Choice D is correct. The equation of a line in the xy-plane can be written as $y=mx+b$, where $m$ represents the slope of the line and $\left(0,b\right)$ represents the y-intercept of the line. It's given that the slope of the line is $\frac{1}{9}$. It follows that $m=\frac{1}{9}$. It's also given that the line passes through the point $\left(0,14\right)$. It follows that $b=14$. Substituting $\frac{1}{9}$ for $m$ and $14$ for $b$ in $y=mx+b$ yields $y=\frac{1}{9}x+14$. Thus, the equation $y=\frac{1}{9}x+14$ represents this line.

Choice A is incorrect. This equation represents a line with a slope of $-\frac{1}{9}$ and a y-intercept of $\left(0,-14\right)$.

Choice B is incorrect. This equation represents a line with a slope of $-\frac{1}{9}$ and a y-intercept of $\left(0,14\right)$.

Choice C is incorrect. This equation represents a line with a slope of $\frac{1}{9}$ and a y-intercept of $\left(0,-14\right)$.