If in two circles, arcs of the same length subtend angles $60^{\circ}$ and $75^{\circ}$ at the centre, find the ratio of their radii.

Vedclass pdf generator app on play store
Vedclass iOS app on app store

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 $

Similar Questions

If $x = \sec \,\phi - \tan \phi ,y = {\rm{cosec}}\phi + \cot \phi ,$ then

The circular wire of diameter $10\,cm$ is cut and placed along the circumference of a circle of diameter $1\, metre.$ The angle subtended by the wire at the centre of the circle is equal to

The value of $k$, for which ${(\cos x + \sin x)^2} + k\,\sin x\cos x - 1 = 0$ is an identity, is

If $5\tan \theta = 4,$ then $\frac{{5\sin \theta - 3\cos \theta }}{{5\sin \theta + 2\cos \theta }} = $

If $x = \sec \theta + \tan \theta ,$ then $x + \frac{1}{x} = $