Help ASAP!!!

[mathsfailure]

Desperately need help with these two questions!

1. Find the value(s) of a such that the line with equation y = x is tangent to the parabola with equation y = x2 + ax+ 1.
2. Find the equation of the straight line(s) which pass through the point (1,−2) and is (are) tangent to the parabola with equation y = x2.

[VanillaRice]

Welcome to ATARNotes

I'm not going to give you the answer, but I will try to get you there:
1) A line acting as a tangent for a parabola means that the two graphs will intersect once. We can find the relevant x-value(s) for intersection by equating x = x² + ax +1. This is essentially another quadratic equation. We know that quadratic equations can have either no, one or two solutions. In our case, we only want the one solution. How can we distinguish the values of a that make this true?

2) I've provided some advice for this exact question elsewhere. I've quoted it below:

I'll try get you started: The equation of a line can be written in the form y = mx + c. Since we know that this line passes through (1, -2), we can sub this point in, giving c = -m-2. We can therefore rewrite the equation of our line as y = mx - m - 2. Since the question asks us to find the equation of the line, our end goal is to find suitable values for m, and substitute these into the equation for our line. We know that the line and parabola intersect (once, as a tangent), so we can solve mx - m - 2 = x². Note that since the line is tangent to the parabola, we are looking for one solution. Where can we go from here, to eventually give us the equation of the line? (Hint: Discriminant)

Hope this helps Post if you get stuck.

[mathsfailure]

Thank you!