Discussion Forum

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

Answer:

Explanation:

 Parametric  equations  of given circle is $x=12 \cos \theta$, $y=12 \sin \theta$

 $[\because $ Parametric equation $x^{2}+y^{2}=r^{2}$ is $x=r \cos \theta$,$y=r\sin \theta$]

 Now, coordinates  of point A are given by

 $x= 12 \cos \frac{\pi}{3}, y=12 \sin \frac{\pi}{3}$

$\Rightarrow$   $x=12.\frac{1}{2},y=12 .\frac{\sqrt{3}}{2}$

$\Rightarrow$    $x=6; y=6 \sqrt{3}$

ie,        A=$(6,6 \sqrt{3})$

 and coordinates of point B are given by

    $x=12\cos \frac{\pi}{6},y=12 \sin \frac{\pi}{6}$

$\Rightarrow$     $x=12. \frac{\sqrt{3}}{2}.y=12 . \frac{1}{2}$

$\Rightarrow$      $x=6\sqrt{3},y=6$

i.e, $B=(6 \sqrt{3},6)$

 Clearly  , length  of chord

$AB=\sqrt{6(\sqrt{3}-6)^{2}+(6-6\sqrt{3})^{2}}$

    $=\sqrt{2\times6^{2}+(\sqrt{3}-1)^{2}}$

     $=6\sqrt{2}(\sqrt{3}-1)$

=$6(\sqrt{6}-\sqrt{2})$