SOLUTION: for what k will the line y=x+k be the tangent to the hyperbola x^2-4y^2=48?

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Question 1024077: for what k will the line y=x+k be the tangent to the hyperbola x^2-4y^2=48?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = y-k.
==> x%5E2-4y%5E2+=+%28y-k%29%5E2-4y%5E2+=+-3y%5E2+-+2ky+%2B+k%5E2+=+48
==> 3y%5E2+%2B+2ky+%2B+48+-+k%5E2+=+0
For tangency, the value of the discriminant b%5E2+-+4ac must be 0:
==> %282k%29%5E2+-4%283%29%2848-k%5E2%29+=+0
==> 4k%5E2+-+12%2848-k%5E2%29+=+0
==> 4k%5E2+-+144+=+0 after simplifying a little further...
==> k%5E2+=+36, or k = 6, -6.
Thus the lines y = x+6 and y = x - 6 are tangent to the hyperbola.