Multiple choice

Let $AC$ be an $Arc$ of circle subtending a right angle at the centre. The point $B$ divides $Arc\ AC$ in ratio $1 : 2$. If $\overline {OA}=\overline {a}$ and $\overline {OB}=\overline {b}$ then, Find $\overline {OC}$ in terms of $\overline {a}$ and $\overline {b}$

  1. $2\overline {b}+\overline {a}\sqrt{3}$
  2. $2\overline {b}-\overline {a}\sqrt{3}$
  3. $-2\overline {b}-\overline {a}\sqrt{3}$
  4. $3\overline {b}+\overline {a}\sqrt{2}$
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B Correct answer
Explanation

Arc AC subtends 90 degrees. B divides AC in 1:2, so angle AOB = 30 degrees and angle BOC = 60 degrees. Using vector rotation or geometry, OC can be expressed as 2b - a*sqrt(3) based on the geometry of the points on the circle.

AI explanation

Placing the center at the origin with OA along the x-axis, the vector OC equals r times the cosine of 90 degrees plus r times the sine of 90 degrees, which simplifies to just r. Since B divides the 90 degree arc in a 1:2 ratio, the angle for OB is 30 degrees, so vector OB equals r times the cosine of 30 degrees plus r times the sine of 30 degrees. Solving this system for r yields r equals 2b minus a multiplied by the square root of 3.

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