If 0 < θ < 90°, 3b cosecθ = a secθ and 3a secθ - b cosecθ = 8, then the value of 9b2 + a2 is:
1. 6
2. 8
3. 9
4. 7
1. 6
2. 8
3. 9
4. 7
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Correct Answer – Option 3 : 9
Given:
3b cosecθ = a secθ
3a secθ – b cosecθ = 8
Formula used:
sin2θ + cos2θ = 1
Calculation:
∵ 3b cosecθ = a secθ
⇒ 9b cosecθ = 3a secθ ——(1)
∵ 3a secθ – b cosecθ = 8
⇒ 9b cosecθ – b cosecθ = 8 ——(From 1)
⇒ 8b cosecθ = 8
⇒ b cosecθ = 1
⇒ b = 1/cosecθ ——(2)
⇒ b = sinθ ——(3)
∵ 3a secθ – b cosecθ = 8
⇒ 3a secθ – (cosecθ/cosecθ) = 8
⇒ 3a secθ – 1 = 8
⇒ 3a secθ = 9
⇒ a secθ = 3
⇒ a = 3/secθ
⇒ a = 3cosθ ——(4)
Putting the values of ‘a’ and ‘b’ from (3) and (4) in 9b2 + a2
= 9 (sinθ)2 + (3cosθ)2
= 9 sin2θ + 9 cos2θ
= 9(sin2θ + cos2θ)
= 9 × 1
= 9