If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Solution:

Let the radii of the circles be r₁ and r₂ be the radii of the two circles.

Let the arc of length l₁ subtend angles 60° at the centre and the arc of the length l₂ be 75° at the centre.

⇒ 60° = π/3 radian and 75° = 5π/12 radian using θ = l / r.

As we know that in a circle or radius r, the arc length l subtend at an angle θ radian at the centre, then θ = l / r or l = r θ

Therefore,

l = r₁ × π/3 and l = r₂ × 5π/12

⇒ r₁ × π/3  = r₂ × 5π/12

On solving them taking r on one side, we get

⇒ r₁ / r₂ = 5π/12 × 3/π 

⇒ r₁ / r₂ = 5/4

NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.1 Question 6

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Summary:

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, then the ratio of the radii of the two circles is 5 : 4