If a chord joining the points P (a sec θ , a tan θ ) & Q (a sec φ , a tan φ ) on the hyperbola x 2 − y 2 = a 2 is a normal to it at P, then show that tan φ = tan θ (4 sec 2 θ − 1) .
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
The equation of normal at p is
x cos θ + y cot θ = 2a solving this with the hyperbola
x 2 – y 2 = a 2 we have
– y 2 = a 2
y 2 (cosec 2 θ – 1) – 4ay sec θ cosec θ + 4a 2 sec 2 θ – a 2 = 0
i.e. product of roots y 1 y 2 = 
Now, y 1 = a tan θ y 2 = a tan φ
so a 2 tan θ tan φ = a 2 (4 sec 2 θ – 1) tan 2 θ
tan φ = (tan θ ) 4(sec 2 θ – 1)
──────────────────────────────────────────────────────────────────────────────────────────
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems