1.

If `(tanx)/2=(tany)/3=(tanz)/5,x+y+z=pi` and `tan^2x+tan^2y+tan^2z=(38)/K` then K=_________`

Answer» Aas `x+y+z = pi`,
`:. tanx+tany+tanz = tanxtanytanz->(1)`
Let `tanx/2 = tany/3 = tanz/5 = k`
`=>tanx = 2k, tany = 3k, tanz = 5k`
Putting these values in (1),
`2k+3k+5k = 2k(3k)(5k)`
`=>10k = 30k^3`
`=>k^2 = 1/3`
`=>k = 1/sqrt3`
Here, we will not take `k = -1/sqrt3` as it will make `x+y+z` negative.
Now, `tanx = 2/sqrt3 ,tany= 3/sqrt3, tanz = 5/sqrt3`
Now, `tan^2x+tan^2y+tan^2z = 38/K`
`=>4/3+9/3+25/3 = 38/K`
`=>38/3 = 38/K`
`=>K = 3`


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