Question

If one of the slopes of the pair of lines $a{x}^2+2hxy+b{y}^2=0$ is $n$ times the other then

• A. $4(n+1{)}^{2}ab=nab$
• B. $4h^2=(n+1{)}^{2}ab$
• C. $4n{h}^2=(n+1{)}^{2}ab$
• D. $4ab=(n+1{)}^{2}h$

Answer 1

Answer: (B)

Let m be the slope of the lines $a{x}^2+2hxy+b{y}^2=0$, then according to question , other slope will be nm.
$\therefore m+nm=\frac{-2h}{b}$
$\Rightarrow m\left(1+n\right)=\frac{-2h}{b}...\left(i\right)$
and$m\cdot nm=\frac{a}{b}$
$\Rightarrow n{m}^2=\frac{a}{b}$
$\Rightarrow m=\pm \sqrt{\frac{a}{bn}}...\left(ii\right)$
$\therefore$ From Eq. (i), we get
$\pm \sqrt{\frac{a}{bn}}\left(1+n\right)=\frac{-2h}{b}$
On squaring both sides, we get
$\frac{a}{bn}{\left(1+n\right)}^2=\frac{4h^2}{b^2}$
$\Rightarrow 4h^{2}n=ab{\left(1+n\right)}^2$