Manipal Entrance Test 2014 Math Paper

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

dx

dy

(x−x1 )
Put y=0, then
x=x1+y1

dy

dx

y12=2x1x
=2x1 (x1+y1

dy

dx

)
⇒ y12=2x12+2x1y1

dy

dx

⇒

dy

dx

=

y12−2x12

2x1y1

=

( y1 x1 )2−2

2 ( y1 x1 )

⇒

dy

dx

=

( y x )2−2

2y x

Put y = vx
⇒

dy

dx

=v+x

dv

dx

We have, v+

xdv

dx

=

v2−2

2v

⇒ x.

dv

dx

=−

(2+v2 )

2v

⇒

2v.dv

2+v2

+

dx

x

=0
On integrating both sides, wo get
log‌ (2+v2 )+log|x|=log‌C
⇒ log‌ (2+v2 )|x|=log‌C
⇒ |x| (2+

y2

x2

)=C
∵ passes through (2,1) then, 2 (2+

1

4

)=C
⇒ C=

9

2

∴

|x| (2x2+y2 )

x2

=

9

2

⇒ 2x2+y2=

9

2

|x|
⇒ 4x2+2y2−9|x|=0