VITEEE 2010 Paper

Let P(x1,y1) be a point on the curve
y2=4‌ax ...(i)
On differentiating y2=4ax w.r.t. 'x', we get
2y

dy

dx

=4a
⇒ (

dy

dx

)(x1,y1)=

2a

y1

Thus, the equation of normal at (x1,y1) is
y−y1=

y1

2a

(x−x1)
⇒ y1x+2‌ay=y1(x1+2a) ...(ii)
But lx+my=1 ...(iii)
is also a normal.
Therefore, coefficients of eqs. (ii) and (iii), must be proportional.
i.e.,

y1

l

=

2a

m

=

y1(x1+2a)

1

⇒y1=

2al

m

and x1=

1

l

−2a
Putting these values of x1‌and‌y1 in eq. (i), we get
(

2al

m

)2=4a(

1

l

−2a)
⇒

4a2l2

m2

=

4a−8a2l

l

⇒ al3=m2−2am2l⇒al3+2alm2=m2