A light house is such that its angle of elevation from a point due north is  θ  and from a point due west is  ϕ .
Also the distance between the points is 'i'
If the height of the lighthouse is  i   tan   θ   tan   ϕ k   tan 2 θ   +   n   tan 2 ϕ  . Find k and n

Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

Here height of light house  =  BD = i tan θ tan ϕk tan2 θ + n tan2 ϕ ( Given ) and distance between both points  =  AC  = i 

We know : tan θ=Opposite Adjacent

So, in triangle BCD we get


tan θ=BDBCtan θ=i tan θ tan ϕ k tan2 θ + n tan2 ϕ BCtan θ=i tan θ tan ϕ BCk tan2 θ + n tan2 ϕ BC=i tan ϕ k tan2 θ + n tan2 ϕ         --- ( 1 ) 

And in triangle ABD we get


tan ϕ=BDABtan ϕ=i tan θ tan ϕ k tan2 θ + n tan2 ϕ ABtan ϕ=i tan θ tan ϕ ABk tan2 θ + n tan2 ϕ AB=i tan θk tan2 θ + n tan2 ϕ         --- ( 2 ) 

And we apply Pythagoras theorem in triangle ABC and get

AC2  =  AB2 +  BC2  , Now we substitute values from equation 1 and 2 and as given distance between two points is " i " .

i2 = i tan θk tan2 θ + n tan2 ϕ 2 + i tan ϕk tan2 θ + n tan2 ϕ 2i2 =i2 tan2 θ k tan2 θ + n tan2 ϕ  + i2 tan2 ϕ k tan2 θ + n tan2 ϕ i2 =i2 tan2 θ + i2 tan2 ϕ k tan2 θ + n tan2 ϕ i2 =i2 tan2 θ +tan2 ϕ k tan2 θ + n tan2 ϕ 1= tan2 θ +tan2 ϕ k tan2 θ + n tan2 ϕ 
We can see that above equation only be satisfied is k = n = 1 .

Therefore ,

k = 1  and n = 1                                                                                    ( Ans )
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  • -1
k =10 n=7 
  • 1
Yes,k=10,n=7
  • 0
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