Question

If the slope of one of the lines represented by ax + 2xy + by - is three times the other then
prove that 3-4ab.​

Answer 1

Answer:

Let l

1

x+m

1

y=0 & l

2

x+m

2

y=0 be the lines represented by ax

2

+2hxy+by

2

=0

⇒(l

1

x+m

1

y)(l

2

x+m

2

y)=ax

2

+2hxy+by

2

⇒l

1

l

2

=a; m

1

m

2

=b; l

1

m

2

+l

2

m

1

=2h

Now, slope of l

1

x+m

1

y=0=

m

1

−l

1

& slope of l

2

x+m

2

y=

m

2

−l

2

According to questions -

⇒

m

1

−l

1

=n.

m

2

−l

2

l

1

m

2

=n.m

1

l

2

l

1

m

2

+l

2

m

1

=(1+n)l

1

m

2

2h=(1+n)l

1

m

2

Also, l

1

l

2

=m

1

m

2

=ab

⇒l

1

m

2

.n.l

1

m

2

=ab

⇒n(l

1

m

2

)

2

=ab

⇒l

1

m

2

=

n

ab

⇒2h=(1+n)

n

ab

required condition.

Hence, solved.