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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).
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