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

• Correct answer
  A

  $6(\sqrt{6}-\sqrt{2})$
• B

  $6(\sqrt{6}-\sqrt{3})$
• C

  $\sqrt{2}(\sqrt{3}-1)$
• D

  $6(\sqrt{3}-1)$

Solution

Parametric equations of given circle is $x=12 \cos \theta, y=12 \sin \theta$ $\left[\because\right.$ Parametric equation of $x^2+y^2=r^2$ is $x=r \cos \theta, y=r \sin \theta$ ] Now, coordinates of point $A$ are given by $$ \begin{aligned} & x=12 \cos \frac{\pi}{3}, y=12 \sin \frac{\pi}{3} \\ & \Rightarrow \quad x=12 \cdot \frac{1}{2}, y=12 \cdot \frac{\sqrt{3}}{2} \\ & \Rightarrow \quad x=6 ; y=6 \sqrt{3} \end{aligned} $$ $$ \text { i.e. } \quad A \equiv(6,6 \sqrt{3}) $$
and coordinates of point $B$ are given by $$ \begin{array}{ll} & x=12 \cos \frac{\pi}{6}, y=12 \sin \frac{\pi}{6} \\ \Rightarrow \quad & x=12 \cdot \frac{\sqrt{3}}{2} \cdot y=12 \cdot \frac{1}{2} \\ \Rightarrow \quad & x=6 \sqrt{3}, y=6 \\ \text { i.e. } \quad & B=(6 \sqrt{3}, 6) \end{array} $$ i.e. $\quad B \equiv(6 \sqrt{3}, 6)$ Clearly, length of chord $$ \begin{aligned} A B & =\sqrt{(6 \sqrt{3}-6)^2+(6-6 \sqrt{3})^2} \\ & =\sqrt{2 \times 6^2(\sqrt{3}-1)^2} \\ & =6 \sqrt{2}(\sqrt{3}-1) \\ & =6(\sqrt{6}-\sqrt{2}) \end{aligned} $$

Hence, the correct answer is option: 'A'.