Final answer:
To find the number of horizontal or vertical tangent lines of the curve xy²=2xy, we need to find the critical points by setting the derivative dy/dx equal to zero or undefined, respectively.
Step-by-step explanation:
The curve xy²=2xy represents a relation between x and y values that satisfies the equation. To find the horizontal or vertical tangent lines, we need to find the critical points of the curve. A critical point occurs when the derivative of the equation equals zero. Taking the derivative of xy²=2xy with respect to x gives us y²+2xy(dy/dx)=2y. Equating this to zero and solving for dy/dx will give us the slope of the tangent lines at the critical points. However, since we are only interested in horizontal or vertical tangent lines, we need to set dy/dx equal to zero or undefined, respectively.