TS EAMCET 9-Sep-2020 Shift 2 Solved Paper

The equation of the hyperbola is
‌

x2

a2

−‌

y2

b2

=1
The equation of the normal at a point
P(a‌sec‌θ,b‌tan‌θ) is given ‌

by

tan‌θ

+‌

ax

sec‌θ

=a2+b2
⇒‌‌b‌cot‌θ‌y+a‌cos‌θ‌x−(a2+b2)=0...(i)
The equation of normal at a point θ(a‌sec‌φ,b‌tan‌φ) is given
‌

by

tan‌φ

+‌

ax

sec‌φ

=a2+b2
⇒b‌cot‌φ‌y+a‌cos‌φ‌x−(a2+b2)=0...(ii)
Solving Eqs. (i) and (ii), we get
‌

x

−(a2+b2)b‌cot‌θ+(a2+b2)b‌cot‌φ

=‌

1

b‌cot‌θ‌a‌cos‌φ−b‌cot‌φ‌a‌cos‌θ

⇒‌

−y

a(a2+b2)[cos‌θ−cos‌φ)

⇒y=

−(a2+b2)

b

(∵θ+φ=‌

π

2

)
⇒‌‌y=‌

−(a2+b2)

b

[‌

cos‌θ−sin‌θ

cos‌θ−sin‌θ

]
⇒‌‌y=−(‌

a2+b2

b

)
As (h,k) is the point of intersection, So,
k=−(‌

a2+b2

b

)