If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles 1 1 cos 7 and sec1 (7) at the centre respectively, then the distance between these chords, is :
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Answer:
If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and subtend angles 1 1 cos 7 and sec1 (7) at the centre respectively, then the distance between these chords, is 8/√7
Step-by-step explanation:
distance between chords
= Radius Cos(subtended angle/2) by chord1 + Radius Cos(subtended angle/2) by chord 2
Radius = Diameter /2 = 4/2 = 2
= 2 Cos (θ₁/2) + 2Cos(θ₂/2)
θ₁ = Cos⁻¹(1/7) => Cosθ₁ = 1/7
θ₂ = Sec⁻¹7 => Secθ₂ = 7 => 1/Cosθ₂ = 7 => Cosθ₂ = 1/7
Applying Cos2θ = 2Cos²θ - 1 => Cos²θ = (1 + cos2θ)/2
Putting θ = θ₁/2
Cos²(θ₁/2) = ( 1 + 1/7)/2 = 4/4
=> Cos(θ₁/2) = 2/√7
Putting θ = θ₂/2
Cos²(θ₂/2) = ( 1 + 1/7)/2 = 4/7
=> Cos(θ₁/2) = 2/√7
2 Cos (θ₁/2) + 2Cos(θ₂/2) = 2 *2/√7 + 2 * 2/√7
= 4/√7 + 4/√7
= 8/√7
Step-by-step explanation:
distance between chords
= Radius Cos(subtended angle/2) by chord1 + Radius Cos(subtended angle/2) by chord 2
Radius = Diameter /2 = 4/2 = 2
= 2 Cos (0₁1/2) + 2Cos(0₂/2)
0₁ = Cos-¹(1/7) => Cos0₁ = 1/7
0₂ = Sec-¹7 => Sec0₂ = 7 => 1/Cos0₂ = 7 => Cos0₂ = 1/7
Applying Cos20 = 2Cos²0 - 1=> Cos²0 = (1 + cos20)/2
Putting 0₁/2 =
Cos (0,₁/2) = (1+1/7)/2 = 4/4
=> Cos(0₁/2) = 2/√7
Putting 0 = = 0₂/2
Cos (0₂/2) = (1+1/7)/2 = 4/7
=> Cos(0₁/2) = 2/√7
2 Cos (0₁/2) +2Cos(0₂/2) = 2 *2/√7 + 2* 2/√7
= 4/√7 + 4/√7
= 8/√7