Let P (a sec θ , b tan θ ) and Q (a sec φ , b tan φ ) where θ + φ = π /2, be two points on the hyperbola x 2 /a 2 – y 2 /b 2 = 1. If (h, k) is the point of intersection of normals at P and Q, then k is equal to -
Text Solution
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Equation of the tangent at P (a sec θ , b tan θ ) is
sec θ – 
tan θ = 1.
Therefore equation of the normal at P is
y – b tan θ = – 
sin θ (x – a sec θ )
⇒ ax + b cosec θ y = (a 2 + b 2 ) sec θ …(1)
Similarly the equation of the normal at
Q (a sec φ , b sec φ ) is
ax + b cosec φ y = (a 2 + b 2 ) sec φ … (2)
Subtracting (2) from (1) we get y = 
.
So that k = y = 
[ θ + φ = π /2]
= 
= – 
.
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