Question

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Answer 1

Answer:

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Let two lines y=m

1

,x and y=m

2

x passess through origin.

⇒y−m

1

x=0 and y−m

2

x=0

⇒(y−m

1

x)(y−m

2

x)=0

∴y

2

−(m

1

+m

2

)xy+m

1

m

2

x

2

=0; Where m

1

and m

2

are slopes of lines.

∴y

2

−(sum of slopes)xy+(product of slopes)x

2

=0

It represents pair of straight lines passing through origin.

Now,

ax

2

+2hxy+by

2

=0

⇒y

2

+

b

2h

xy+

b

a

x

2

=0

⇒m

1

+m

2

=

b

−2h

and m

1

m

2

=

b

a

⇒A.M≥G.M

⇒

2

m

1

+m

2

≥

m

1

m

2

⇒(m

1

+m

2

)

2

≥4m

1

m

2

⇒

b

2

4h

2

≥

b

4a

⇒h

2

≥ab

∴h

2

−ab≥0