Question

eliminate theta x= asectheta+btantheta y=bsec theta + atan theta​

Answer 1

Answer:

Squaring both sides of a tan θ + b sec θ = x we get,

(a tan θ + b sec θ)2 = x2 , …………….. (A)

Now, squaring both sides of a tan θ + b sec θ = y we get,

(a tan θ + b sec θ)2 = y2, …………….. (B)

Now subtract (B) from (A) we get,

x2 - y2 = (a tan θ + b sec θ)2 - (a tan θ + b sec θ) 2

⇒ x2 - y2 = (a 2 tan2 θ + b 2 sec2 θ + 2ab tan θ sec θ) - (a2 tan2 θ + b2 sec2 θ + 2ab tan θ sec θ)

⇒ x2 - y2 = b2 tan2 θ + a2 sec2 θ + 2ba tan θ sec θ - a2 tan2 θ - b2 sec2 θ - 2ab tan θ sec θ

⇒ x2 - y2 = b2 tan2 θ - a2 tan2 θ + a2 sec2 θ - b2 sec2 θ

⇒ x2 - y2 = tan2 θ (b2 – a2) + sec 2 θ (a2 - b2)

⇒ x2 - y2 = - tan2 θ (a2 - b2) + sec 2 θ (a2 - qb2) ⇒ x2 - y2 = sec2 θ (a2 - b2) - tan2 θ (a2 - qb2)

⇒ x2 - y2 = (a2 – b2) (sec2 θ - tan2 θ)

⇒ x2 - y2 = (a2 – b2)(1), [Since sec 2 θ - tan2 θ = 1]

⇒ x2 - y2 = a2 – b2..