Question

If the arcs of same length in two circles subtend angles of 60°
and 75° at their centres. Find the ratio of their radii.​

Answer 1

SOLUTION :-

Let,

Radius of first circle = $\sf r_{1}$

Radius of second circle = $\sf r_{2}$

We know that,

$\boxed{\bf l = \theta \ r}$

$\bullet\sf \ Length \ of \ arc \ of \ first \ circle$

  $\longrightarrow \sf l = r_{1}\times \theta_{1} \\\\\longrightarrow l = r_{1}\times 60^{\circ} \\\\\longrightarrow l = r_{1}\times 60\times \dfrac{\pi}{180}\\\\\longrightarrow \bf l = r_{1}\dfrac{\pi}{3}$

$\bullet \sf \ Length \ of \ arc \ of \ second \ circle$

  $\longrightarrow \sf l = r_{2} \times \theta_{2} \\\\\longrightarrow l = r_{2}\times 75^{\circ} \\\\\longrightarrow l = r_{2}\times 75 \times \dfrac{\pi}{180}\\\\\longrightarrow \bf l = r_{2} \dfrac{5 \pi}{12}$

According to the question,

Length of arc of first circle = Length of arc of second circle

$\longrightarrow \sf r_{1} \dfrac{\pi}{3}= r_{2}\dfrac{5\pi}{12}\\\\\longrightarrow \dfrac{r_{1}}{r_{2}} = \dfrac{5\pi}{12}\times \dfrac{3}{\pi} \\\\\longrightarrow \dfrac{r_{1}}{r_{2}}= \dfrac{15}{12} \\\\\longrightarrow \bf \dfrac{r_{1}}{r_{2}}= \dfrac{5}{4}$

Ratio of their radii = 5 : 4