I WILL GIVE is an ellipse. The line y=mx+2 is tangent to the ellipse. Determine the value of (). Express your answer as an integer or as a common or improper fraction.

9514 1404 393

  m^2 = 35/9

Step-by-step explanation:

The line will be tangent to the ellipse where there is one point of intersection between the line and the ellipse. We can find that case by looking at the discriminant of the quadratic.

  x^2 +9(mx +2)^2 = 1

  x^2 +9(m^2x^2 +4mx +4) = 1

  (9m^2 +1)x^2+36mx +35 = 0 . . . . subtract 1, collect terms

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Then the discriminant is ...

  (36m)^2 -4(9m^2 +1)(35)

We want it to be zero, so we have ...

  1296m^2 -1260m^2 -140 = 0 . . . . eliminate parentheses

  36m^2 = 140 . . . . . . . . . . . . . . . add 140

  m^2 = 140/36 . . . . . . . . . . . . divide by 36

  m^2 = 35/9

translate k to n and reflect across the line containing jk.

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step-by-step explanation: