UPSC CDS 2 2023 Math Paper

Concept: For equal arc lengths, the product of radius and angle (in degrees) is constant because arc length formula is $l = \frac{\pi r \theta}{180^{\circ}}$.Explanation:Let $r_1$ be the radius of circle $C_1$ and $r_2$ be the radius of circle $C_2$.Angle subtended at the centre by the arc of $C_1$ is $\theta_1 = 60^{\circ}$.Angle subtended at the centre by the arc of $C_2$ is $\theta_2 = 75^{\circ}$.Both arcs have equal length, so $l_1 = l_2$.Arc length formula: $l = \frac{2 \pi r \theta}{360^{\circ}} = \frac{\pi r \theta}{180^{\circ}}$.Equating the two arc lengths: $\frac{\pi r_1 \cdot 60^{\circ}}{180^{\circ}} = \frac{\pi r_2 \cdot 75^{\circ}}{180^{\circ}}$.Cancel the common factor $\frac{\pi}{180^{\circ}}$: $60 \, r_1 = 75 \, r_2$.Divide both sides by 15: $4 r_1 = 5 r_2$.Thus $\frac{r_1}{r_2} = \frac{5}{4}$.Hence the ratio of the radius of $C_1$ to the radius of $C_2$ is $5 : 4$.Alternatively, for equal arcs, radius is inversely proportional to the angle; so $r_1 : r_2 = \theta_2 : \theta_1 = 75^{\circ} : 60^{\circ} = 5 : 4$.Answer: B. $5 : 4$.