Question

Find the point of intersection of line x/a+y/b=1,x/b+y/a=1(a≠±b)

Answer 1

Dear Student,
Please find below the solution to the asked query:

We have,
x/a+y/b=1

⇒bx+ay=ab...(i)
x/b+y/a=1
⇒ax+by=ab .....(ii)

(i)×a−(ii)×b

abx+a^2.y−abx−b^2.y
=a^2b−ab^2

⇒(a^2−b^2)y
=ab(a−b)

⇒(a−b)(a+b)y=ab(a−b)

⇒y=ab(a−b) / (a−b)(a+b)
⇒y=ab / (a+b)

Now putting this value in (ii), we get
ax+ab^2(a+b)=ab
⇒a(a+b)x+ab^2(a+b)=ab
⇒a(a+b)x+ab^2
=a^2b+ab^2

⇒a(a+b)x
=a^2b+ab^2−ab^2

⇒x=a^2.b / a(a+b)
⇒x=ab/(a+b)
So, (ab/a+b,ab/a+b) is the point of intersection of the given pair of lines(a)

As, the point
(ab/(a+b),ab/(a+b))
satisfies x−y=0
So, lies on the line x−y=0(b)
As, (ab/a+b,ab/a+b) also satisfies the line (x+y)

(a+b)=2ab

So, (ab /a+b,ab/a+b)
also lies on the line
(x+y)(a+b)=2ab(c)
As, (ab/a+b,ab/a+b) also satisfies the line (lx+my)(a+b)=(l+m)ab
So, also lies on the line
(lx+my)(a+b)=(l+m)ab

Hence, the correct option is (d).

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