The gradint of one of the line of ax² +2hxy+by² =0 is twice that of the other, then

1) h²=ab

2) h=a+b

3) 8h²=9ab

4) 9h²=8ab

also please explain what is gradient ?

Gradient is the vector pointing in the direction of the greatest rate of increase of the scaler.

in this case,  (slope 1)2 = (slope 2)

so, 2(-h + (h^2 - ab)^(1/2))/b =(-h - (h^2 - ab)^(1/2))/b

solving this we get,

8h^2 = 9ab

Here,gradient refers to the slope of the line.

And further,in pair of straight lines,m₁+m₂=-2h/b and m₁m₂=a/b,where m₁ and m₂ are the slopes of the two lines represented by the equation ax²+2hxy+by²=0.

Let m₁=m,then m₂=2m₁=2m.

Now,m+2m=-2h/b

→3m==-2h/b

→m=-2h/3b...............................(1)

Also,(m)(2m)=a/b

→2m²=a/b

→m²=a/2b

→(-2h/3b)²=a/2b..............................from (1)

→4h²/9b² =a/2b

→8h²=9ab

So,option (3) is the correct answer.

Hi Pooja,

Let the line be of the form y=mx.

So the eqn becomes, a+2hm+bm^2 = 0.

One root (ie the gradient) is twice of the other.

So m+2m = -2h/b = 3m

And m*2m = a/b = 2m^2

So a/b = 2(-2h/3b)^2

So 8h^2 = 9ab.

Option (3).

Hope that helps.

Best Regards,

Ashwin (IIT Madras).