VITEEE 2014 Paper

The equation of any normal
to

x2

a2

−

y2

b2

= 1 is ax cos Φ + by cot Φ = a2+b2
⇒ ax cos Φ + by cot Φ - (a2+b2) = ... (i)
The straight line lx +my-n = 0 will be normal to the hyperbola

x2

a2

−

y2

b2

= 1 , then eq. (i) and lx + my-n =
0 represent the same line,
∴

acosΦ

l

=

bcotΦ

m

=

a2+b2

n

⇒ sec Φ =

na

l(a2+b2)

and tan Φ =

nb

m(a+b2)

n2a2

l2(a2+b2)

−

n2b2

m2(a2+b2)2

= 1
[Since sec2 Φ - tan2 Φ = 1]
⇒

a2/

l2

−

b2

m2

=

(a2+b2)2

n2

But given equation of normal is

a2

l2

−

b2

m2

=

(a2+b2)2

k

∴ k = n2