Question

If the slope of one of the lines represented by $a{x}^2-6xy+y^2=0$ is the square of the other, then the value of $a$ is

• A. $-27$ or $8$
• B. $-3$ or $2$
• C. $-64$ or $27$
• D. $-4$ or $3$

Answer 1

The correct option is A $-27$ or $8$

The above equation of pair lines depicts two lines passing through the origin.
Hence, let the lines be $y=mx$ and $y=kx$.
Therefore
$(y-mx)(y-kx)=0$ implies
$y^2-(m+k)x+mk{x}^2=0$
Comparing with the above equation gives us
$m+k=6$
Now it is given that $k=m^2$
Hence
$m+m^2=6$
$m^2+m-6=0$
$(m+3)(m-2)=0$
$m=-3$ and $m=2$
Hence the slopes can be
$(-3,9)$ or $(2,4)$.
Thus the values of a can be
$-27$ or $8$.