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 \[b=\sqrt{2}\,a\] |
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