Question

If $a$ is an integer and $m_1,m_2,m_3$ are the slopes of all three straight lines represented by the equation $y^3+(2a+5)x{y}^2-6x^{2}y-2a{x}^3=0$ which are also integers then which of the following can hold good?

• A. $a+\sum _{i=1}^3{m}_i=-1$
• B. $a+\sum _{i=1}^3{m}_i=-5$
• C. $a\prod _{i=1}^3{m}_i=0$
• D. $a\prod _{i=1}^3{m}_i=32$

Answer 1

Answer: (A), (B), (C), (D)

putting $y=mx$ in the equation
$m^3+(2a+5)m^2-6m-2a=0$
$m^3+5m^2-6m+2a\left(m^2-1\right)=0$
$m\left(m^2+5m-6\right)+2a\left(m^2-1\right)=0$
$m(m+6)(m-1)+2a(m-1)(m+1)=0$
$m(m+6)+2a(m+1)=0\text{ or }m=1$
$2a=\frac{-m(m+6)}{(m+1)}=-m-5+\frac{5}{m+1}$
for a to be an integer, we must have
$m+1=\pm 1\text{ or }\pm 5$
$m=0\text{ or }-2\text{ or }4\text{ or }-6$
for $m=0a=0$ and other values of $m$ are $m(m+6)=0$
$m=0,-6\text{ and }1$
for $m=-2a=-4$ and other values of $m$ are $m^2-6m-8m-8=0$
$m^2-2m-8=0$
$(m-4)(m+2)=0$
$m=-2,4\text{ and }1$
$\therefore a=0,m_1=1,m_2=0,m_3=-6$
$\text{ and }a=-4,m_1=1,m_2=-2,m_3=4$