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Question

If A(θ) and B(ϕ) are the parametric ends of a focal chord of x2144y225=1, then the maximum value of tanθ2tanϕ2 is 


Solution

Given: x2144y225=1 a=12,b=5 
e2=1+b2a2e2=1+25144e=1312

The endpoints of the focal chord are A=(asecθ,btanθ) and B=(asecϕ,btanϕ)
Then, the equation of chord AB is 
xacos(θϕ2)ybsin(θ+ϕ2)=cos(θ+ϕ2)

It passes through (±ae,0), so
cosθ+ϕ2cosθϕ2=±e

When cosθ+ϕ2cosθϕ2=e 
Applying componendo and dividendo, we get
tanθ2tanϕ2=e1e+1tanθ2tanϕ2=131211312+1=125
When cosθ+ϕ2cosθϕ2=e
Applying componendo and dividendo, we get
tanθ2tanϕ2=e1e+1tanθ2tanϕ2=131211312+1=25
Therefore, the maximum value of tanθ2tanϕ2 is 25

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