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TMUA 2017 · Paper 1 · Question 9 of 20

TMUA 2017 Paper 1 Question 9

Coordinate geometry — Circle equation · inscribed regular hexagon. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Coordinate geometryCircle equation · inscribed regular hexagon8 options
A circle has equation x2 + y2 − 18x − 22y + 178 = 0 A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle.

What is the area of the hexagon?

  1. A6
  2. B6√3
  3. C18
  4. D18√3
  5. E36
  6. F36√3
  7. G48
  8. H48√3
Show the answer and worked solution
answer · F
  1. A6
  2. B6√3
  3. C18
  4. D18√3
  5. E36
  6. F36√3
  7. G48
  8. H48√3
Complete the square: (x−9)2 + (y−11)2 = 81 + 121 − 178 = 24, so r2 = 24. A regular hexagon inscribed in a circle of radius r splits into six equilateral triangles of side r, so its area is 6 × √34r2 = 3√32r2. With r2 = 24 that is 36√3. Note that only r2 is needed, so there is no reason to simplify √24.