Question

If one of the slopes of the pair of lines $ax^2 + 2hxy + by^2 = 0$ is $n$ times the other then

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

Answer 1

Answer: (B)

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