Question

find point of intersection of lines x=a and y=b​

Answer 1

Answer:

we have,

a

x

+

b

y

=1and

b

x

+

a

y

=1

⇒bx+ay−ab=0⟶(1)

and,ax+by−ab=0⟶(2)

Equation of line through their point of intersection will be given by:

(bx+ay−ab)+k(ax+by−ab)=0

where k is constant

⇒x(b+ak)+y(a+bk)−ab(k−1)=0

Line meet x-axis at A, hence for A, y=0

⇒x(b+ak)−ab(k−1)=0

⇒x=

(b+ak)

ab(k−1)

so, coordinate of A is (

(b+ak)

ab(k−1)

,0)

Line meets y-axis at B, hence for B, x=0

⇒y(a+bk)−ab(k−1)=0

⇒y=

(a+bk)

ab(k−1)

so, coordinate of B is (0,

(a+bk)

ab(k−1)

)

Let (h,m) be midpoint of AB, hence

h=

2

(b+ak)

ab(k−1)

+0

⇒k=

a

b

(

b−2h

2h+a

)⟶(3)

and,m=

2

0+

a+bk

ab(k−1)

⇒k=

b

a

(

a−2m

2m+b

)⟶(4)

From (3) and (4) we get

⇒

a

b

(

b−2h

2h+a

)=

b

a

(

a−2m

2m+b

)

To get equation of locus we take h→xandm→y

⇒

a

b

(

b−2x

2x+a

)=

b

a

(

a−2y

2y+b

)

⇒2xy(a+b)=ab(x+y)