Question

The $L_{1}$ and $L_{2}$ denoted by $3 x^{2}+10 x y+8 y^{2}+14 x+22 y+15$ $=0$ intersect at the point $P$ and have gradients $m _{1}$ and $m _{2}$ respectively. The acute angles between them is $\theta$. Which of the following relations hold good?

• A. $m _{1}+ m _{2}=5 / 4$
• B. $m _{1} m _{2}=3 / 8$
• C. acute angle between $L_{1}$ and $L_{2}$ is $\sin ^{-1} \frac{2}{5 \sqrt{5}}$
• D. sum of the abscissa and ordinate of the point $P$ is $-1$.

Answer 1

Answer: (B), (C), (D)

$f ( x , y )=( x +2 y +3)(3 x +4 y +5) $
$m _{1}+ m _{2}=-\frac{2 h }{ b }=-\frac{10}{8}=\frac{-5}{4} $
$m _{1} \,m _{2}=\frac{ a }{ b }=\frac{3}{8}$
Also $L _{1}$ and $L _{2}$ intersect at $(1,-2)$
$\therefore $ Sum of abscissa and ordinate $=-1$
$\tan \theta=\frac{2 \sqrt{ h ^{2}- ab }}{ a + b }=\frac{2 \sqrt{25-24}}{11}=\frac{2}{11} $
$\sin \theta=\frac{2}{5 \sqrt{5}} $
$\Rightarrow \theta=\sin ^{-1}\left(\frac{2}{5 \sqrt{5}}\right)$