Question

If the equation $a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3=0$ $(a,b,c,d\ne 0)$ represents three coincident lines, then

• A. $a=c$
• B. $b=d$
• C. $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$
• D. $ac=bd$

Answer 1

The correct option is C $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$

$a{x}^3+3b{x}^{2}y+3cx{y}^2+d{y}^3=0$ ---(1) represent three coincident line
Let $y=mx$ is that line
$(y-mx{)}^3=0$
$y^3+3m^{2}xy-3y^2\left(mx\right)-m^3{n}^3=0$
$m^3{x}^3+3mn{y}^2-3m^{2}xy-y^3=0$ ---(2)
Compare 1 & 2,
$\frac{a}{m^3}=\frac{b}{m}=\frac{c}{-m^2}=\frac{d}{-1}$
$\therefore \frac{a}{b}=\frac{b}{c}=\frac{c}{d}$