if ax^3+by^3+cx^2y+dxy^2=0 represent three distinct straight lines,such that each line bisects the angle between other two then prove that 3b+c=0

2 years ago

Share

Answers : (1)

                    

Let the lines be


 


y-mix = 0 where i=1,2,3


 


As one line is equally inclined to the other two, we impose the following condition.


(m1-m3)/(1+m1m3) = -(m2-m3)/(1+m2m3)


Simplify to get,


 


(m1+m2+m3)-3m3-m3(m1m2+m2m3+m3m1)+3m1m2m3 = 0       ...........(i)


 


Similarly applying the condition on the other two pairs, we get,


 


(m1+m2+m3)-3m2-m2(m1m2+m2m3+m3m1)+3m1m2m= 0         ............(ii)


and


(m1+m2+m3)-3m1-m1(m1m2+m2m3+m3m1)+3m1m2m= 0          .......... (iii)


 


Adding equations (i),(ii) and (iii), we get,


 


-(m1m2+m2m3+m3m1)(m1+m2+m3) + 9m1m2m3 = 0                 ............(iv)


 


Putting y=mx in the equation of combined equation,


 










ax3+by3+cx2y+dxy2=0




we get,


 


bm3 + dm2 + cm + a = 0


By theory of equations,


 


m1+m2+m3 = -d/b


m1m2+m2m3+m3m1 = c/b


m1m2m3 = -a/b


 


 


Putting the above relations in equation (iv),


 


-(c/b)(-d/b) + 9(-a/b) = 0


 


cd-9ab=0


 


This is the condition that i get. No idea where i went wrong.

2 years ago

Post Your Answer

More Questions On Analytical Geometry

Ask Experts

Have any Question? Ask Experts
Post Question
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!!
Click Here for details
find the eccentricity of the ellipse whose foci are (2,4) and (14,9) and touches the x axis
 
 
since ellipse touches the x-axis, therefore x-axis is a tangent to the ellipse. The product of the perpendicular distance is equal to square of the semi minor. Therefore b 2 =9*4=36 distance...
  img
Sunil Raikwar 5 months ago
if a parabola whose length of latus rectum is 4a touches the coordinate axes the find the locus of its focus
 
 
please check the attached file
  img
Sunil Raikwar 5 months ago
 
Let (h,k) be the focus of the hyperbola, since it touches the axis tangent at the vertex should be and the equation of its directrix should be and the perpendicular distance from direcrtrix...
 
sunil raikwar 5 months ago
 
Hi Pranjal, There is slight technical issue. Please post these questions again in analytical Geometry. We will upload the answers for the same. askIITians Faculty
 
sunil raikwar 5 months ago
Area of a quadrilateral is 3/4root3. The radius of circle circumscribing the quad. is 1. if AB=1 BD=root3 then BC.CD=?
 
 
Hello Student, clearly Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians Faculty
  img
Arun Kumar one month ago
Cos(2pi/2n+1) +cos(4pi/2n+1)+cos(6pi/2na1).......+cos(2npi/2n+1)
 
 
Hello Student, let x=2pi/(2n+1) => cosx+cos2x+cos3x+cos4x.......cosn=sin((n+1)x/2)cos(nx/2)/sin(x/2) replace x by its value Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians...
  img
Arun Kumar one month ago
 
Dear student, You may use the following result: Let Now multiply both members with Using the identity we have Then Substitute with Then Regards Sumit
  img
Sumit Majumdar one month ago
the third term of a geometric progression is 4 product of first five term is?
 
 
Hello Student, Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians Faculty
  img
Arun Kumar 21 days ago
2,5,8,.........100 terms 3,5,7...........100 terms Find the sum of all the common terms of both the AP’s?
 
 
Hello Student, Common terms will come at interval of 6 => 5,11,17 will go untill 201 Now apply the AP formulae n/2*(2a+(n-1)*d) Arun Kumar Btech IIT delhi Askiitians Faculty
  img
Arun Kumar 20 days ago
 
Also refer to this link http://www.askiitians.com/forums/Algebra/2-5-8-100-terms-3-5-7-100-terms_104140.htm
  img
Arun Kumar 20 days ago
View all Questions »