question_answer

Two chords of lengths a m and b m subtend angles \[60{}^\circ \]and \[90{}^\circ \] at the centre of the circle, respectively. Which of the following is true?                                                                              [SSC (CGL) Mains 2014]

A) \[b=\sqrt{2}a\]  

B) \[a=\sqrt{2}b\]

C) \[a=2b\]                       

D) \[b=2a\]

Correct Answer: A

Solution :

In \[\Delta AOB,\]

\[AO=BO=r\][radius of circle]

\[{{b}^{2}}={{r}^{2}}+{{r}^{2}}\]

\[b=\sqrt{2{{r}^{2}}}\]

\[\Rightarrow \]   \[b=\sqrt{2}\,r\]                         (i)

In \[\Delta COD,\]

\[\angle COD=60{}^\circ \]

Then,    \[\angle OCD=\angle ODC\]

\[=180{}^\circ -\angle COD\]

\[=180{}^\circ -60{}^\circ \]

\[=120{}^\circ \]

Also, \[\angle OCD=\angle ODC=\]Angle opposite to equal sides.

\[\therefore \]\[\angle OCD=\angle ODC=60{}^\circ \]

So, \[\Delta COD\] is equilateral and \[r=a\]           ... (ii)

From Eqs. (i) and (ii), we get \[b=\sqrt{2}\,a\]