Question

If the slope of one of the lines represented by the equation $a{x}^2+2hxy+b{y}^2=0$ be square of the other, then the value of $\frac{a+b}{h}+\frac{8h^2}{ab}$ is

• A. $4$
• B. $-6$
• C. $6$
• D. $-4$

Answer 1

The correct option is C $6$

Let the equation of the lines by $y=m_{1}x$ and $y=m_{2}x$
then $m_2=m_1^2....\left(i\right)$
Also, $m_1+m_2=-\frac{2h}{b}$
$\Rightarrow m_1+m_{1}2=\frac{-2h}{b}....\left(ii\right)$
and $m_1{m}_2=\frac{a}{b}\Rightarrow m_1^3=\frac{a}{b}....\left(iii\right)$
Taking cube of both the sides of the Eq. (ii), we get
$m_1^3+m_1^6+3m_1^3(m_1+m_1^2=-\frac{8h^3}{b^3}$
Again, from Eq.s (ii) ad (iii), we get
$\left(\frac{a}{b}\right)+{\left(\frac{a}{b}\right)}^2+3\left(\frac{a}{b}\right)\left(\frac{-2h}{b}\right)=\frac{-8h^2}{b^3}$
$\Rightarrow a^{2}b+a{b}^2+8h^2=6abh$
On dividing both sides by $abh$, we get
$\frac{a+b}{h}+\frac{8h^2}{ab}=6$.