Correct answer: C. 9
To find the points of intersection between the parabola and the line, we set their equations equal to each other:
`f(x) = 2x + c`
`-x² + 8x = 2x + c`
For the line and the parabola to intersect at exactly one point, this equation must have exactly one real solution. To determine this, we first rearrange the equation into the standard quadratic form `ax² + bx + k = 0` (using `k` for the constant to avoid confusion with the `c` in the problem).
`0 = x² - 8x + 2x + c`
`0 = x² - 6x + c`
A quadratic equation has exactly one real solution when its discriminant, `b² - 4ac`, is equal to zero. In the equation `x² - 6x + c = 0`, the coefficients are `a=1`, `b=-6`, and the constant term is `c`.
Set the discriminant to zero and solve for `c`:
`(-6)² - 4(1)(c) = 0`
`36 - 4c = 0`
`36 = 4c`
`c = 9`
Therefore, the correct value of `c` is 9.
Choice A is incorrect. This is the x-coordinate of the point of tangency, found by solving `x² - 6x + 9 = 0`, which gives `(x-3)² = 0`, so `x=3`. The question asks for the value of `c`.
Choice B is incorrect. This value results from assuming the line must pass through the vertex of the parabola. The vertex of `f(x)` is at `x = -8/(2 · -1) = 4`, and `f(4) = 16`. If the line `y=2x+c` passed through `(4,16)`, then `16 = 2(4) + c`, which means `c = 8`. However, a line passing through the vertex is not necessarily tangent to the parabola.
Choice D is incorrect. This value results from misremembering the discriminant formula, for instance by solving `b² - c = 0` instead of `b² - 4ac = 0`. This would give `(-6)² - c = 0`, so `c=36`.