Manipal Entrance Test 2013 Math Paper

Variable line is

x

a

+

y

b

=1 .....(i)
Any line perpendicular to (i) and passing through the origin will be

x

b

−

y

a

=0 ...(ii)
Now foot of the perpendicular from the origin to line (i) is the point of intersection (i) and (ii)
Let it be P(α, β), then

α

a

+

β

b

=1 .....(iii)
and

α

a

−

β

b

=1 .....(iv)
Squarring and adding Eqs. (m) and (iv), we get
α2(

1

a2

+

1

b2

)+β2(

1

b2

+

1

a2

)=1
∴(α2+β2)

1

c2

=1
Hence, the locus of P(alpha,beta) is x2+y2=c2 which is a circle