Answer
The tangent is $$y=-\frac{5}{4}x+6$$ and the normal is $$y=\frac{4}{5}x-\frac{11}{5}$$
Work Step by Step
$$x+\sqrt{xy}=6\hspace{2cm}(4,1)$$
a) Find the derivative of the function: $$1+\frac{1}{2\sqrt{xy}}(xy)'=0$$ $$1+\frac{y+xy'}{2\sqrt{xy}}=0$$ $$1+\frac{y}{2\sqrt{xy}}+\frac{xy'}{2\sqrt{xy}}=0$$ $$\frac{2\sqrt{xy}+y}{2\sqrt{xy}}+\frac{xy'}{2\sqrt{xy}}=0$$ $$y'=-\frac{\frac{2\sqrt{xy}+y}{2\sqrt{xy}}}{\frac{x}{2\sqrt{xy}}}=-\frac{2\sqrt{xy}+y}{x}$$
b) The slope of the tangent at $(4,1)$ is $$y'=-\frac{2\sqrt{4\times1}+1}{4}=-\frac{2\times2+1}{4}$$ $$y'=-\frac{5}{4}$$
The tangent to the curve at $(4,1)$ is $$y-1=-\frac{5}{4}(x-4)$$ $$y-1=-\frac{5}{4}x+5$$ $$y=-\frac{5}{4}x+6$$
c) Since the normal is the perpendicular line to the tangent at a point, the product of their slopes equals $-1$.
Therefore, if we call the slope of the normal at $(4,1)$ $s$, we would have $$s\times\Big(-\frac{5}{4}\Big)=-1$$ $$s=\frac{4}{5}$$
The normal to the curve at $(4,1)$ is $$y-1=\frac{4}{5}(x-4)$$ $$y-1=\frac{4}{5}x-\frac{16}{5}$$ $$y=\frac{4}{5}x-\frac{11}{5}$$
