SOLUTION: Please help me with those problems. Thanks!!!!! 1)Graph x2 + 2xy + y2 = 1. 2)If the parabola de fined by y = ax^2 + 6 is tangent to the line y = x, then calculate the constant a.

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Question 477245: Please help me with those problems. Thanks!!!!!
1)Graph x2 + 2xy + y2 = 1.
2)If the parabola de fined by y = ax^2 + 6 is tangent to the line y = x, then calculate the constant a. (A line is tangent to a parabola if it touches the parabola at one point, but is otherwise always "outside" the parabola.)
3)Find a complex number whose square equals i.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
1) x%5E2+%2B+2xy+%2B+y%5E2+=+1
<==> %28x%2By%29%5E2+-+1+=+0
<==> (x+y-1)(x+y + 1) = 0.
The graph is the union of the two lines y = -x + 1 and y = -x - 1.
graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+-x-1%2C+-x+%2B1%29
2) Solve for the system y = x and y+=+ax%5E2+%2B+6 such that there is only one point of intersection.
==> Let +x+=+ax%5E2+%2B+6 ==> ax%5E2+-+x+%2B+6+=+0
For there to be only one point of intersection, the discriminant has to be equal to 0.
==> %28-1%29%5E2+-+4%2Aa%2A6+=+0, or 1- 24a = 0, or a+=+1%2F24.
3) Let x+=+e%5E%28i%2Atheta%29. Since i+=+e%5E%28%281%2F2%29%2Ai%2Api%29,
we have %28e%5E%28i%2Atheta%29%29%5E2+=+e%5E%282i%2Atheta%29+=+e%5E%28%281%2F2%29%2Ai%2Api%29
Hence we can let 2i%2Atheta+=+%28i%2F2%29%2Api, or
theta+=+pi%2F4.
thus a complex number whose square is i is