UPSC NDA Math Model Paper 8

Given, the equation of lines are

x

a

+

y

b

=1 and

x

b

+

y

a

=1
The equation to the straight line joining the origin to the point of intersection of the lines

x

a

+

y

b

=1 and

x

b

+

y

a

=1 is
(

x

a

+

y

b

−1)+λ(

x

b

+

y

a

−1)=0 ........(1)
which is passing through the origin.
⇒(

0

a

+

0

b

−1)−λ(

0

b

+

0

a

−1)=0
⇒λ=−1
Equation of line becomes,
⇒(

x

a

+

y

b

−1)+(−1)(

x

b

+

y

a

−1)=0
⇒(

x

a

+

y

b

−1)−(

x

b

+

y

a

−1)=0
⇒

x

a

+

y

b

−1−

x

b

−

y

a

+1=0
⇒x(

1

a

−

1

b

)−y(

1

a

−

1

b

)=0
⇒x−y=0
Hence, the equation to the straight line joining the origin to the point of intersection of the lines

x

a

+

y

b

=1 and

x

b

+

y

a

=1 is x−y=0.