Multiple choice

Circle drawn with a focal chord of the parabola $y ^ { 2 } = 4 x + y + 4$ as diameter will always touch the line $16 x + k = 0 ,$ where $k$ equals to :

  1. $8$
  2. $16$
  3. $33$
  4. $36$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The parabola is y^2 - y = 4x + 4. Completing the square: (y - 1/2)^2 = 4x + 4 + 1/4 = 4x + 17/4 = 4(x + 17/16). The focus is at (h+a, k) = (-17/16 + 1, 1/2) = (-1/16, 1/2). A circle with a focal chord as diameter touches the directrix. The directrix is x = h - a = -17/16 - 1 = -33/16. The line is 16x + k = 0, or x = -k/16. Comparing x = -33/16 and x = -k/16, k = 33.

AI explanation

Rewriting the parabola y^2 - y = 4x + 4, we complete the square to get (y - 1/2)^2 = 4x + 17/4, or Y^2 = 4X where Y = y - 1/2 and X = x + 17/16. The focus of this parabola is at X = 1, giving x = -1/16. The directrix is the line X = -1, meaning x + 17/16 = -1, which simplifies to 16x + 33 = 0. A circle drawn with a focal chord as its diameter will always touch the directrix, so the line is 16x + 33 = 0 and k equals 33.

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