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)
⇒
(∵ φ + θ = π / 2) Create a free account to view solution
View Solution FreeMore Hyperbola Questions
The equation to the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyper bola xy = c2 is -...The value of m for which y = mx + 6 is a tangent to the hyperbola = 1 is-...Locus of the middle points of the parallel chords with gradient m of the rectangular hyperbola xy = c2 is...If radii of director circle of = 1 and = 1 are 2r and r respectively and ee and eh be the eccentricities of the ellipse ...The equation of hyperbola with vertices (-6,0) and (6,0) and semi latus rectum , is given by -...