If the Arcs of the Same Length in Two Circles Subtend Angles 65° and 110° at the Centre, Then the Ratio of the Radii of the Circles is

Question:

If the arcs of the same length in two circles subtend angles \(65^\circ\) and \(110^\circ\) at the centre, then the ratio of the radii of the circles is

(a) \(22 : 13\)

(b) \(11 : 13\)

(c) \(22 : 15\)

(d) \(21 : 13\)

Solution

We know that arc length is given by:

\[ l = r\theta \]

Since the arcs are of the same length,

\[ r_1\theta_1 = r_2\theta_2 \]

Given:

\[ \theta_1 = 65^\circ,\qquad \theta_2 = 110^\circ \]

So,

\[ r_1 \times 65 = r_2 \times 110 \]

\[ \frac{r_1}{r_2} = \frac{110}{65} \]

\[ = \frac{22}{13} \]

Therefore, the ratio of the radii is:

\[ 22 : 13 \]

Hence, the correct option is:

(a) \(22 : 13\)