Question : Two chords of lengths $a$ metre and $b$ metre subtend angles $60^{\circ}$ and $90^{\circ}$ at the centre of the circle, respectively. Which of the following is true?
Option 1: $b=\sqrt{2}a$
Option 2: $a=\sqrt{2}b$
Option 3: $a=2b$
Option 4: $b=2a$
Correct Answer: $b=\sqrt{2}a$
Solution :
The length of a chord in a circle,
$=2×r×\sin \left(\frac{\theta}{2}\right)$
Where $r$ is the radius of the circle and $\theta$ is the angle subtended at the centre by the chord.
Given that the lengths of the chords are $a$ and $b$ meters, they subtend angles of $60^{\circ}$ and $90^{\circ}$ at the centre of the circle respectively.
$ \therefore a = 2 ×r ×\sin \left(\frac{60^{\circ}}{2}\right) = 2× r ×\sin30^{\circ}$...(i)
$ \therefore b = 2×r× \sin \left(\frac{90^{\circ}}{2}\right)= 2×r× \sin45^{\circ}$...(ii)
From equation (i) and (ii),
$⇒\frac{a}{2 ×\sin30^{\circ}} = \frac{b}{2× \sin45^{\circ}}$
$⇒a = b \times \frac{\sin30^{\circ}}{\sin45^{\circ}}$
$⇒a=b×\frac{1}{\sqrt{2}}$
$⇒b=\sqrt{2}a$
Hence, the correct answer is $b=\sqrt{2}a$.
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