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Question

The area of the triangle formed by the tangents from the point (3,2) to the hyperbola x29y2=9 and the chord of contact with respect to the point (3,2) is 


Solution

x29y2=9
x29y2=1
Any point on the hyperbola is (3secθ,tanθ)
Tangent at this point is 
3secθ x9tanθ y=9
secθ x3tanθ y=3
This tangent passes through (3,2)
3secθ 6tanθ =3
secθ 2tanθ =1
sec2θ=1+4tanθ+4tan2θ
1+tan2θ=1+4tanθ+4tan2θ
3tan2θ+4tanθ=0
tanθ=0,  tanθ=43  
for tanθ=0, secθ=1
and for tanθ=43, secθ=53 (  point R lies in third quadrant)
A=12∣ ∣ ∣ ∣3213015431∣ ∣ ∣ ∣
=123(0+43)2(3+5)+1(4+0)
=12|4164|
=8

Mathematics

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