Question

$\huge\mathfrak\red {Question!}$
If the arcs of the same lengths in two circles subtend angles 65° and 110° at the centre, find the ratio of their radii.​

Answer 1

$\huge\mathcal\red{\underline{\underline{Solution}}}$

☞ $\sf{θ_1 = 65°}$

☞ $\sf{θ_1 = \frac{π}{180°} × 65°}$

☞ $\sf{θ_1 = \frac{π}{36} × 13}$

☞ $\sf{θ_1 = \frac{13π}{36} radians}$

Similarly

☞ $\sf{θ_2 = 110°}$

☞ $\sf{θ_2 = \frac{π}{180°} × 110°}$

☞ $\sf{θ_2 = \frac{π}{36} × 22}$

☞ $\sf{θ_2 = \frac{22π}{36} radians}$

Now,

$\bf\green{Using\: formula}$

$\large\boxed{θ = \frac{l}{r}}$

☞ $\sf{θ_1 = \frac{l_1}{r_1}}$

☞ $\sf{l_1 = θ_1 × r_1}$

☞ $\sf{l_1 = \frac{13π}{36} × r_1}$......(1)

☞ $\sf{θ_2 = \frac{l_2}{r_2}}$

☞ $\sf{l_2 = θ_2 × r_2}$

☞ $\sf{l_2 = \frac{22π}{36} × r_2}$......(2)

$\bf\green{From\: equation\: (1)\: and\: (2),\: we\: get}$

➮ $\sf{\frac{13π}{36} × r_1 = \frac{22π}{36} × r_2}$

➮ $\sf{13π × r_1 = 22π × r_2}$

➮ $\sf{\frac{r_1}{r_2} = \frac{22π}{13π}}$

➮ $\sf{\frac{r_1}{r_2} = \frac{22}{13}}$

☃️$\huge{\boxed{\sf{r_1 : r_2 = 22 : 13}}}$

Answer 2

WKT ,

l = r Θ

let the radius of two circle be r1 and r2 .

length of arc of 1st circle

l = r1 Θ

= r1 × 65°

= r1 × 65° × π/180°

= r1 × 13π/36

length of arc 2nd circle

l = r2 Θ

= r2 × 110°

= r2 × 110° × π/180°

= r2 × 11π/10

Given That ,

r1 × 13π/36 = r2 × 11π/10

r1/r2 = 11π/36 × 13π/36

r1/r2 = 22π/13π

r1/r2 = 22/13

Therefore , r1 : r2 = 22 : 13

therefore the ratio of radius is 22 : 13 //

Hence you got the answer