Manipal Entrance Test 2014 Math Paper

Equation of line is

x

a

+

y

b

=1 ....(i)
Let the foot of the perpendicular drawn from theorigin to the line be P(x1,y2). since, OP⊥AB
∴ slope of OP× slope of AB=−1
⇒ y1x1× b−a=−1 ⇒by1=ax1 ...(ii)
Since, P lies on the line AB, so

x1

a

+

y1

b

=1 ...(iii)
⇒ bx1+ay1=ab
From (ii) and (iii), we get
x1=

ab2

a2+b2

and y1=

a2b

a2+b2

Now, x12+y12=(

ab2

a2+b2

)2 + (

a2b

a2+b2

)2
=

a2b4

(a2+b2 )2

+

a4b2

(a2+b2 )2

=

a2b2

(a2+b2 )2

(a2+b2 )
=

a2b2

a2+b2

=

1

1 a2 + 1 b2

⇒x12+y12=

1

c2

(∵

1

a2

+

1

b2

=

1

c2

)
Thus, the locus of P(x1,y1) is x2+y2=c2, which is a circle.