HELP PLEASE!! I'm lost with this problem? Consider the parabola y=x^2...?

Question

A.) Write the equation to the tangent line to the prabola y=x^2 @ (a,a^2).
B.) Find all tangent lines to the parabola y = x^2 that pass through (0,-2).

----Now let (c,d) be a point which may or may not lie on the parabola.
C.) Make a conjecture as to how many tangent lines of the parabola pass through a given point (c,d). How does the answer depend on the point (c,d)?
D.) Give conditions on c and d which tells us whether we have exactly 0, 1 or 2 tangent lines through (c,d).
E.) What do these conditions tell us about the location of the point (c,d) with respect to the parabola?

Some help please? I am lost with this problem and it's due tomorrow.

Answer 1

Well, I won't write the equations for you, but I can answer the other questions. If that helps.

B: Since (0,-2) is directly below the "point" of the parabola, there will be exactly 2 tangent lines passing through it. They will be reflections about the y axis.

C: Most points will also have 2 tangent lines passing through them. See next answer for what determines how many
D: If (c,d) lies INSIDE the parabola, there will be 0 tangent lines passing through it.
If (c,d) lies ON the parabola, there will be 1 tangent line passing through it.
If (c,d) lies OUTSIDE the parabola (as in B: above), there will be 2 tangent lines passing through it.

Answer 2

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