Question

The differential equation is dy/dx = (y - 3)/ (x-3). Select the following options that are true.

Answer 1

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.