HyperbolaHard
Question
Let P (a sec θ, b tan θ) and Q(a sec φ, b tanφ), where θ + φ = 
, be two point on the hyperbola 
= 1. If (h, k) is the point of the intersection of the notmals at P and Q, then k is equal to
, be two point on the hyperbola = 1. If (h, k) is the point of the intersection of the notmals at P and Q, then k is equal to Options
A.
B.
C.
D.
Solution
Firstly we obtain the slope of notmal to = 1. at (a sec θ, b tan θ) On differentiating w. r. t. x, we get
⇒ 
Slope, for notmal at the point (a sec θ, b tan θ) will be
∴ Equation of normal at (a sec θ, b tan θ) is
y - b tan θ = -
sin θ (x - a sec θ)
⇒ (asin θ)x + by = (a2 + b2 tan θ)
⇒ ax + bcosec θ = (a2 + b2) sec θ .......(i)
Similarly, equation of normal to
at (a sec φ ,b tan φ) is
ax + b ycosec φ = (a2 + b2) sec φ .......(ii)
On subtracting Eqs (ii) from (i), we get
b(cosec θ - cosec φ) y = (a2 + b2) (sec θ - sec φ)
⇒
(∵ φ + θ = π / 2)
= 1. at (a sec θ, b tan θ) On differentiating w. r. t. x, we get ⇒ Slope, for notmal at the point (a sec θ, b tan θ) will be
∴ Equation of normal at (a sec θ, b tan θ) is
y - b tan θ = -
sin θ (x - a sec θ) ⇒ (asin θ)x + by = (a2 + b2 tan θ)
⇒ ax + bcosec θ = (a2 + b2) sec θ .......(i)
Similarly, equation of normal to
at (a sec φ ,b tan φ) is ax + b ycosec φ = (a2 + b2) sec φ .......(ii)
On subtracting Eqs (ii) from (i), we get
b(cosec θ - cosec φ) y = (a2 + b2) (sec θ - sec φ)
⇒
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