Question

Two circles \(C_1\) and \(C_2\) have radii 18 and 12 units, respectively. If an arc of length \( \ell \) of \(C_1\) subtends an angle 80° at the centre, then the angle subtended by an arc of same length \( \ell \) of \(C_2\) at the centre is:

• A. 90°
• B. 100°
• C. 110°
• D. 120°
• E. 135°

Hint:

To find the angle subtended by an arc at the center of a circle, use the arc length formula relating angle, arc length, and radius. Adjustments in radius directly influence the subtended angle when arc length remains constant.

Answer 1

The Correct Option is D

The angle \(\theta\) subtended by an arc of a circle at the center is given by the formula: \[ \theta = \frac{\ell}{r} \times \frac{180}{\pi} \] where \( \ell \) is the length of the arc and \( r \) is the radius of the circle. For circle \(C_1\) with radius \( r_1 = 18 \) units: \[ 80^\circ = \frac{\ell}{18} \times \frac{180}{\pi} \] Solving for \( \ell \): \[ \ell = \frac{80 \pi}{180} \times 18 = 8\pi \] Now, using this \( \ell \) for circle \(C_2\) with radius \( r_2 = 12 \) units: \[ \theta_2 = \frac{8\pi}{12} \times \frac{180}{\pi} \] \[ \theta_2 = \frac{8 \times 180}{12} = 120^\circ \]