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?

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

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

C) \[a=2b\]

D) \[b=2a\]

Correct Answer: A

Solution :

[a] In \[\Delta AOB,\] AO = BO = r [radius of circle] \[{{b}^{2}}={{r}^{2}}+{{r}^{2}}\] \[b=\sqrt{2{{r}^{2}}}\] \[b=\sqrt{2}\,r\]             (i) In \[\Delta COD,\]           \[\angle COD=60{}^\circ \] Then, \[\angle OCD+\angle ODC=180-\angle COD\] \[=180-60=120{}^\circ \] Also \[\angle OCD=\angle OCD=\]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), \[b=r\sqrt{2}\,a\]