Hi, there. I'm happy to help with this.
What you ask is a very sophisticated question ---basically, you are tip-toeing into the beginning concepts of differential calculus here, which is way way at the outer limit of what the GMAT could even conceivably expect.
First of all, I've posted the technical answer to your more general question. The short answer, though is: for parabolas, for ordinary meat-and-potato quadratic functions, a tangent line will never pass through the function at the point of tangency nor will it intersect the function again at another point. The tangent to a parabola will always be on the "outside" of the parabola and never cross into the "inside" part.
I will also say --- certainly all the technical info in the pdf, and even the short answer about the parabola, are, as far as I can determine, beyond anything you will need for the GMAT. This DS question, I would say, is way harder than anything the GMAT would ask.
As for this question:
Prompt:
In the xy-plane, does the line L intersect the graph of y = x^2
A straightforward yes/no prompt.
Statement #1:
Line L passes through (4, -8)
That point is not on the parabola, so a line through that point could intersect the parabola, or it could go in another direction. This statement by itself is
insufficient.
Statement #2:
Line L passes through (-4, 16)
This point is on the parabola. Most lines through this point will intersect the parabola. Line L is a tangent line at that point, which means it would "intersect" at only a single point, but that's still an intersection.. Therefore, this statement by itself is
sufficient.
Answer =
B
Does all this make sense? Let me know if you have any further questions.
Mike
