If `ax^3+by^3+cx^2y+dxy^2=0`, represents three distinct straight lines, such that each line bisects the angle between the other two, then which of the following is true:

Options

A. `3b+c=0`
B. `3a+d=0`
C. `d+3a=0`
D. `b+3c=0`

Answer

A

Explanation

The given equation, being a homogeneous equation of degree 3, represents three lines passing through the origin. Let `m_(1),m_(2),m_(3)` be their slopes. Then `m_(1),m_(2) and m_(3)` are roots the equation
`bm^(3)+dm^(2)+cm+a=0`
`rArr" "m_(1)+m_(2)+m_(3)=-(d)/(b),m_(1)m_(2)+m_(2)m_(3)+m_(3)m_(1)=(c)/(b)`
`and, m_(1)m_(2)m_(3)=-(a)/(b)`
Let `f(m)=bm^(3)+dm^(2)+cm +a`
If the lines represented by the given equation are distinct and each bisects the angle between the other two, then the angle between any two adjacent lines is `(2pi)/(3)` and the cubic equation f(m) = 0 has three disinct real roots.
Now,
f(m) = 0 has three distinct real roots.
`rArr" "f'(m)=0` has two distinct real roots `alpha, beta` such that `f(alpha)f(beta)lt0`.
`rArr" "3bm^(2)+2dm+c=0` has two distinct real roots `alpha,beta` such that `f(alpha)f(beta)lt0`.
`rArr" "4d^(2)-12cgt0 rArr d^(2)-3c gt 0`
If each line bisects the angle between the other two, then
`(m_(1)-m_(2))/(1+m_(1)m_(2))=(m_(2)-m_(3))/(1+m_(2)m_(3))=(m_(3)-m_(1))/(1+m_(1)m_(3))=pmsqrt3`
`rArr" "(1+m_(1)m_(2))/(m_(1)-m_(2))=(1+m_(2)m_(3))/(m_(2)-m_(3))=(1+m_(3)m_(1))/(m_(3)-m_(1))`
`rArr" "3+m_(1)m_(2)+m_(2)m_(3)+m_(3)m_(1)=0`
`rArr" "3+(c)/(b)=0rArr3b+c=0`.
Hence, `3b+c=0 and d^(2)gt3c`