Multiple choice

If $A\left( \cfrac { \pi }{ 3 } \right) ,B\left( \cfrac { \pi }{ 6 } \right) $ are the points on the circle represented in parametric form with centre $(0,0)$ and radius $12$, then the length of the chord $AB$ is

  1. $6\left( \sqrt { 6 } -\sqrt { 2 } \right) $
  2. $6\left( \sqrt { 6 } -\sqrt { 3 } \right) $
  3. $\sqrt { 2 } \left( \sqrt { 3 } -1 \right) $
  4. $6\left( \sqrt { 3 } -1 \right) $
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A Correct answer
Explanation

Points are (R cos theta1, R sin theta1) and (R cos theta2, R sin theta2). R=12, theta1=pi/3, theta2=pi/6. Chord length = sqrt((x2-x1)^2 + (y2-y1)^2) = 2R sin((theta1-theta2)/2) = 24 * sin(pi/12). sin(pi/12) = sin(15) = (sqrt(6)-sqrt(2))/4. Length = 24 * (sqrt(6)-sqrt(2))/4 = 6(sqrt(6)-sqrt(2)).

AI explanation

Using the parametric coordinates, the points are A(6, 6 root 3) and B(6 root 3, 6). The distance formula AB = root((x2 - x1)^2 + (y2 - y1)^2) gives AB = root((6 root 3 - 6)^2 + (6 - 6 root 3)^2), which equals root(2(6 root 3 - 6)^2). Factoring out the 6 yields AB = 6(root 6 - root 2).

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