Statement 1 : If -2h=a+b , then one li...

Statement 1 : If `-2h=a+b ,` then one line of the pair of lines `a x^2+2h x y+b y^2=0` bisects the angle between the coordinate axes in the positive quadrant. Statement 2 : If `a x+y(2h+a)=0` is a factor of `a x^2+2h x y+b y^2=0,` then `b+2h+a=0` .

Both the statements are true but statement 2 is the correct explanation of statement 1.

Both the statements are true but statement 2 is not the correct explanation of statement 1.

Statement 1 is true and statement 2 is false.

Statement 1 is false and statement 2 is true.

Text Solution

Verified by Experts

The correct Answer is:

Put `2h=-(a+b)`in`ax^(2)+2hxy+by^(2)=0`. Then ,
`ax^(2)-(a+b)xy+by^(2)=0`
or `(x-y)(ax-by)=0`
Therefore , one of the lines bisects the angle between the coordinates axes in the positive quadrant. Also , putting
`b=-2h-a` in `ax-by` ,we have
`ax-by=ax-(-2h-a)y=ax+(2h+a)y`
Hence , `ax+(2h+a)y` is a factor of `ax^(2)+2hxy+by^(2)` . However, statement 2 is not the correct explanation of statement 1.

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If one of the pair of lines ax^2+2hxy+by^2=0 bisects the angle between coordinate axes in positive quadrant, then

`a+b=2|h|`

`a+b=-2h`

`a-b=2|h|`

`(a-b)^2=4h^2`

If pairs of straight lines x^2-2pxy-y^2=0 and x^2-2qxy-y^2=0 be such that each pair bisects the angle between the other pair ,then

pq=1

pq=-1

p+q=1

p+q=-1

The angle between the lines 2x - y + 3 = 0 and x + 2y + 3 =0

`30^@`

`45^@`

`60^@`

`90^@`