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
What is the integration of 7^xe^x dx
 
 
Hello Student, Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians Faculty
  img
Arun Kumar 6 months ago
A straight line through p(2,3) inclined at an angle 60 degrees with positive y axis in clockwise direction.The co-ordinates of one of the points on it at a distance 4 units from P is.?
 
 
solution: as shown.. the equn of the line (x – x 1 ) / cos A = ( y – y 1 ) / sin A = R where A = angle of inclination with X-axis R = distance of the point on the line so ( x...
  img
Ajay Verma 6 months ago
The line lx+my+n=0 bisects the angle between a pair of straight lines of which one is px+qy+r=0. Show that the equation to the other line is:- (px+qy+r)(l^2+m^2) -2(lp+mq)(lx+my+n) =0
 
 
Form a family of straight lines passing through the point of intersection oflx+my+n=0 andpx+qy+r=0. (px+qy+r)+µ(lx+my+n)=0 Find the angle betweenlx+my+n=0 andpx+qy+r=0 and equate it to...
  img
Ravi 16 days ago
x^1/3 + y^1/3 + z^1/3 = 0 find (x + y + z)^3
 
 
Hello Student Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians Faculty
  img
Arun Kumar 6 months ago
If p,q are the roots of the quadratic equation x 2 -10rx-11s=0 and r,s are roots of x 2 -10px-11q=0,then find the value of p+q+r+s.
 
 
Hello student, Please find the answer to your question below Givenx 2 -10rx-11s=0........(1) x 2 -10px-11q=0....(2) As p is a root of 1 we get p 2 -10pr-11s=0 and r is a root of 2 so r 2...
  img
SHAIK AASIF AHAMED 5 months ago
 
p, q are the roots of the quadratic equation x 2 - 10rx – 11s = 0 . r, s are the roots of the quadratic equation x 2 – 10px – 11q = 0. The two equations are perfectly symmetric with roots...
 
Satyajit Samal 5 months ago
 
1210
 
PRAMAY 5 months ago
show that of all the rectangles inscribed in a fixed circle ,the square has the maximum area
 
 
Hello Student, Thanks & Regards Arun Kumar Btech, IIT Delhi Askiitians Faculty
  img
Arun Kumar 6 months ago
View all Questions »