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જો બે વર્તુળોમાં સમાન લંબાઈનાં ચાપ કેન્દ્ર આગળ $60^{\circ}$ અને $75^{\circ}$ ના ખૂણા આંતરે, તો તેમની ત્રિજ્યાઓનો ગુણોત્તર શોધો.
$5: 4 $
$5: 4 $
$5: 4 $
$5: 4 $
Solution
Let the radii of the two circles be $r_{1}$ and $r_{2} .$ Let an arc of length $l$ subtend an angle of $60^{\circ}$ at the centre of the circle of radius $r_{1},$ while let an arc of length/subtend an angle of $75^{\circ}$ at the centre of the circle of radius $r_{2}$
Now, $60^{\circ}=\frac{\pi}{3}$ radian and $75^{\circ}=\frac{5 \pi}{12}$ radian
We know that in a circle of radius $r$ unit, if an arc of length $l$ unit subtends an angle $\theta$ radian at the centre then
$\theta=\frac{l}{r}$ or $l=r \theta$
$\therefore l=\frac{r_{1} \pi}{3}$ and $l=\frac{r_{2} 5 \pi}{12}$
$\Rightarrow \frac{r_{1} \pi}{3}=\frac{r_{2} 5 \pi}{12}$
$\Rightarrow r_{1}=\frac{r_{2} 5}{4}$
$\Rightarrow \frac{r_{1}}{r_{2}}=\frac{5}{4}$
Thus, the ratio of the radii is $5: 4 $