Question

The condition that the slope of a line represented by $a{x}^2+2hxy+b{y}^2=0$ is square of the other is

• A. $ab(a+b-6h)+8h^3=0$
• B. $ab(a+b+6h)=8h^2$
• C. $ab(a+b+6h)+8h^3=0$
• D. $ab(a+b-6h)=8h^2$

Answer 1

The correct option is A $ab(a+b-6h)+8h^3=0$

Dividing by $x^2$, $b{m}^2+2hm+a=0$ $\{m=y=slope\}$

$\Rightarrow$ roots of this equation would be , $\alpha ,{\alpha }^2$

$\Rightarrow \alpha +{\alpha }^2=-2h/b$ & ${\alpha }^3=a/b\Rightarrow \alpha =(\frac{a}{b}{)}^{1/3}$

$\Rightarrow (\frac{a}{b}{)}^{1/3}+(\frac{a}{b}{)}^{2/3}=-\frac{2h}{b}$

$\Rightarrow a^{1/3}.b^{2/3}+a^{2/3}{b}^{1/3}=-2h$ __ (1)

$\Rightarrow \left(ab{)}^{1/3}\right(a^{1/3}+b^{1/3})=-2h$

Cubing, $\left(ab\right)(a+b+3(a^{2/3}{b}^{1/3}+a^{1/3}{b}^{2/3}\left)\right)=(-2h{)}^3$

From (1), $\left(ab\right)(a+b+3(-2h\left)\right)=-8h^3$

$\Rightarrow \left(ab\right)(a+b-6h)=-8h^3\Rightarrow \left(A\right)$