a. Write an expression for the slope of the curve at any point (x,y)

I can do this one, I get (-2x-y)/(x+2y)

b. Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel.

Ok this one I’m not so sure about. Do I find the x-intercepts of the curve from the original forumla and plug it into the slope equation, assuming I did it correctly? And if they are equal then the lines are parralell?

c. Find the points on the curve where the lines tangent to the curve are vertical.

Ok for this one I’m confused. Verticle lines have no slope, so how can I use the slope equation to find the points? Wouldn’t they be non-existant?
Please help me on this, it’s driving me nuts.

The ‘x-intercepts’ are where y=0, and so y’ = -2x/x = -2. There are two values of x where y=0, these are ± sqrt(27).

To say a line ‘has no slope’ equals saying ‘has zero slope’. The slope of a vertical line is sometimes said to be ‘undefined’ because the denominator in the derivative is zero. Now, one could opt to argue from this point of view to avoid doing the problem, but doing so only amounts to pedantry and isn’t serious mathematics. On the other hand, in order to accomplish something, simply say ‘the derivative goes toward infinity because the denominator goes toward zero’. And while these two characters are sent of on their merry ways toward infinity and zero, just look for values of (x,y) where x+2y =0.

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I was taught (rather, since I’m still in HS, am being taught) that no slope meant an undefined slope (=vertical line), whereas zero slope meant exactly that - a slope of zero (=horizontal line).
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