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The gradint of one of the line of ax 2 +2hxy+by 2 =0 is twice that of the other, then 1) h 2 =ab 2) h=a+b 3) 8h 2 =9ab 4) 9h 2 =8ab also please explain what is gradient ?


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 ?


Grade:12

3 Answers

Ameya Panse
14 Points
9 years ago

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

Menka Malguri
39 Points
9 years ago

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.

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

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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