Hyperbola

General form of hyperbola is

x2

a2

−

y2

b2

=1 ...(i)
Given equation of hyperbola is 2x2−y2=2
⇒x2−

y2

2

=1 ...(ii)
comparing equation (i) with (ii), we get
a2=1,b2=2
General equation of normal,

a2x

x1

+

b2y

y1

=a2+b2 ...(iii)
Equation of normal at point A(sec‌θ,2‌tan‌θ)

1×x

sec‌θ

+

2×y

2‌tan‌θ

=1+2
⇒

x

sec‌θ

+

y

tan‌θ

=3
⇒

x

sec‌θ

+

y

sin‌θ cos‌θ

=3
⇒

x

sec‌θ

+y‌cos‌θ×cosec‌θ=3
⇒

x

sec‌θ

+

y

sec‌θ

‌cosec‌θ=3
⇒x+y‌cosec‌θ=3‌sec‌θ ...(iv)
Similarly, equation of normal at point B(sec‌ϕ,2‌tan‌ϕ)
x+y‌cosec‌ϕ=3‌sec‌ϕ
⇒x+y‌cosec(

π

2

−θ)=3‌sec(

π

2

−θ)
⇒x+y‌sec‌θ=3‌cosec‌θ ...(v)
subtract equation (v) from (iv), we get
⇒y(cosec‌θ−sec‌θ)=3(sec‌θ−cosec‌θ)
⇒y(cosec‌θ−sec‌θ)=−3(cosec‌θ−sec‌θ)
⇒y=−3
Now, from equation (ii), we get
x2−

y2

2

=1
x2−

9

2

=1
x2=1+

9

2

x=√

11

2

∴(x,y)=(α,β)=(√

11

2

,−3)
∴(2β)2=(2×(−3))2=36