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 $y^3+(2 a+5) x y^2-6 x^2 y-2 a x^3=0$ which are also integers then which of the following can hold good?

• A. $a +\displaystyle\sum_{i=1}^3 m _i=-1 $
• B. $ a +\displaystyle\sum_{i=1}^3 m _i=-5$
• C. $a \displaystyle\prod_{i=1}^3 m _i=0$
• D. $a \displaystyle\prod_{i=1}^3 m _i=32$

Answer 1

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

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