The differential equation is dy/dx = (y - 3)/ (x-3). Select the following options that are true. I. produces a slope field with horizontal tangents at y = 3
II. produces a slope field with vertical tangents at y = 3
III. produces a slope field with rows of parallel segments

-I've narrowed it down to:
A. I only or B. III only

Step-by-step explanation:

The curve produces a slope field with vertical tangents at y = 2.

Step-by-step explanation:

The differential equation  equals the quotient of the quantity x minus 2 and y minus 2.

Hence,

Now, at y = 2,  becomes ∞ and hence the curve y = f(x) with  will have tangents at y = 0 with slopes equal to ∞ i.e. the tangents make 90° angle with the positive x-axis.

Therefore, the curve produces a slope field with vertical tangents at y = 2. (Answer)

Horizontal tangents are such that dy/dx = 0; this happens when

(y - 3) / (x - 3) = 0   →   y - 3 = 0   →   y = 3

so (I) is true.

Vertical tangents are such that their slope is undefined; this happens when the denominator vanishes, or

x - 3 = 0   →   x = 3

so you are right that (II) is not true.

To assess whether (III) is correct, consider what happens at different nearby points (with neither x = 3 nor y = 3) where the y coordinate is kept the same so that the tangent lines occur in the same row. For example,

x = 0, y = 0   →   dy/dx = (-3)/(-3) = 1

x = 1, y = 0   →   dy/dx = (1 - 3)/(-3) = 2/3

x = 2, y = 0   →   dy/dx = (2 - 3)/(-3) = 1/3

The slopes are not the same - they have to be if these tangents are supposed to be parallel - so (III) is not true.

This makes A. (I) only the correct choice.

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