1
advanced-math

Find the value of kk so that the line y=2x+ky = 2x + k is tangent to the parabola y=x2y = x^2.

A

-1

B

1

C

0

D

-2

Correct Answer: A

Choice A is the correct answer. A line is tangent to a parabola if they intersect at exactly one point.

  1. Set equal: x2=2x+kx^2 = 2x + k.
  2. Rearrange: x2−2x−k=0x^2 - 2x - k = 0.
  3. Set discriminant to 0: Δ=(−2)2−4(1)(−k)=0\Delta = (-2)^2 - 4(1)(-k) = 0.
  4. 4+4k=0⇒4k=−4⇒k=−14 + 4k = 0 \Rightarrow 4k = -4 \Rightarrow k = -1.

Choice B is incorrect. Choice C is incorrect. Choice D is incorrect.

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Set 20: Quadratic Equations and Functions 20