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


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)


(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,



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




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
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
Arun Kumar 2 months ago
Angle between pair of Staines x^2+2xy-y^2 is?
Dear student, If we compare this with the standard equation: ax 2 +2hxy+by 2 =0 We get, a =1, h = 1, b = -1 Hence using the formula: we get: Hence, Regards Sumit
Sumit Majumdar 2 months ago
Find the centres and radii of two circles passing through points (7,2) and (10,-1) respectively and both touches the line x=4 ? please solve this question and tell what assumptions should be...
Ans: Hello Student, Please find answer to your question below Let the centre of 1 st circle to be (a, b) Then, radius of circle be: a – 4 Circle is passing through (7, 2) & (4, b)....
Jitender Singh 2 months ago
evaluate the following integral ∫ (2 cos 2x + sec^2 (x)) / sin 2x + tan x – 5 dx
Ans: Hello student, Please find the answer to your question below
Jitender Singh 28 days ago
If a and b are positive numbers , prove that the equation 1/x + (1/x+a) +(1/x+b) =0 has 2 real roots ,one b/w a/3 and 2a/3 and other b/w – 2b/3 and – b/3
Ans: Hello Student, Please find the solution of your question below, -> So equation f(x) has two real roots. You can easily found the value from here.
Jitender Singh 2 months ago
Sum 1+3+4+8+7+13+10+18+........... to 23 terms.
Hello Student, if you see its merged form of two series. 1,4,7,10,13,16...till 12 terms=6(2+11*3) 3,8,13...till 11 terms=11(3+10*5)/2 Now its easy to find. Thanks & Regards Arun Kumar...
Arun Kumar 2 months ago
View all Questions »