If 0 < θ < 90°, 3b cosec θ = a sec θ and 3a sec θ - b cosec θ = 8, then the value of $9b^2 + a^2$ is :

9

Given :-

3b cosec θ = a sec θ  -----(1)

and 3a sec θ - b cosec θ = 8      -----(2)

multiply equation 1 by 3 .

9b cosec θ = 3a sec θ

By putting value of ( 3a sec θ ) into equation 2.

9b cosec θ - b cosec θ = 8

8b cosec θ = 8

b cosec θ = 1

b = sinθ

Put value of ' b ' into equation 1.

3b cosec θ = a sec θ

3 × sinθ . cosec θ = a sec θ 

a = 3 cosθ

Now,

9b² + a²

= 9 ( sin²θ ) + (3 cosθ)²

= 9  ( sin²θ + cos²θ )

= 9       { sin²θ + cos²θ  = 1 }