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If the arcs of the same length in two circles subtend angles $65^{\circ}$ and $110^{\circ}$ at the centres of the circles. Find the ratio of the radii of the circles.

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22 : 13

22 : 11

22 : 10

22 : 15

answer is A.

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Detailed Solution

Assume that
The radius of the two circle =

r_{1} and r_{2}

Their arc of same length = S
Subtend angle =

65^{\circ}

and

110^{\circ}

Now,
We know that if a circle of radius r and a circle of arc length l are subtended at their centres by an angle of x degrees, then we have

l = \frac{x}{360} \times 2 πr

C:UserssoftberryDownloadsP0041 JPEGQ.81.jpg

Hence, we have

l = \frac{65}{360} \times 2 π r_{1} - (1)

And

l = \frac{110}{360} \times 2 π r_{2} - (2)

From equation (1) and (2), we obtain

\frac{65}{360} \times 2 π r_{1} = \frac{110}{360} \times 2 π r_{2}

65 r_{1} = 110 r_{2}

[Multiply by

\frac{360}{2 π}

]

r_{1} = \frac{110}{65} r_{2}

r_{1} = \frac{22}{13} r_{2}

\frac{r_{1}}{r_{2}} = \frac{22}{13}

\Rightarrow r_{1} : r_{2} = 22 : 13

Therefore, the ratio of the radii = 22: 13
Option 1 is Correct.

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