Correct Answer - Option 4 : 99
Formula used:
- (a - b)2 = a2 - 2ab + b2
-
(a + b)2 = a2 + 2ab + b2
-
sec2θ - tan2θ = 1
Calculation:
Given that,
6 + 8 tan θ = sec θ
Multiply with 6 on both sides,
36 + 48 tan θ = 6sec θ ----(1)
Also, we have
8 - 6 tan θ = k sec θ
Multiply with 8 on both sides,
64 - 48 tan θ = 8k sec θ -----(2)
Adding equation (1) & (2), we get
36 + 48 tan θ + 64 - 48 tan θ = 6sec θ + 8k sec θ
⇒ 100 = sec θ(6 + 8k)
⇒ sec θ = 100/(6 + 8k) -----(3)
Since, 6 + 8 tan θ = sec θ
⇒ 6 + 8 tan θ = 100/(6 + 8k)
⇒ 8 tan θ = 100/(6 + 8k) - 6
⇒ 8 tan θ = (100 - 36 - 48k)/(6 + 8k)
⇒ 8 tan θ = 8(8 - 6k)/(6 + 8k)
⇒ tan θ = (8 - 6k)/(6 + 8k) ----(4)
We know that,
sec2θ - tan2θ = 1
Hence, from equations (3) and (4)
[100/(6 + 8k)]2 - [(8 - 6k)/(6 + 8k)]2 = 1
⇒ [1002 - (8 - 6k)2]/(6 + 8k)2 = 1
⇒ 10000 - 64 - 36k2 + 96k = 36 + 64k2 + 96k
⇒ 10000 - 100 = 100k2
⇒ 9900 = 100k2
⇒ k2 = 99
∴ The value of k2 is 99.