Let P (a sec θ, b tan θ) and Q(a sec φ, b tanφ), where θ + φ = , be tw

Question

Let P (a sec &#952;, b tan &#952;) and Q(a sec &#966;, b tan&#966;), where &#952; + &#966; = , 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

Accepted Answer

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)

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