Question

The angle between the pair of lines whose equation is $4x^2+10xy+m{y}^2+5x+10y=0$ is

• A. ${\tan }^{-1}\left(3/8\right)$
• B. ${\tan }^{-1}\left(3/4\right)$
• C. ${\tan }^{-1}\frac{2\sqrt{25-4m}}{(m+4)},m\in R$
• D. none of these

Answer 1

The correct option is C ${\tan }^{-1}\frac{2\sqrt{25-4m}}{(m+4)},m\in R$

General equation of pair of straight line.

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0So,a=4,b=m,h=5,g=\frac{5}{2},f=5,c=0$

Angle

$={\tan }^{-1}\left[\frac{2{\left(h^2-ab\right)}^{\frac{1}{2}}}{\left|a+b\right|}\right]$

$={\tan }^{-1}\left[\frac{2{\left(5^2-4m\right)}^{\frac{1}{2}}}{\left|4+m\right|}\right]$

$={\tan }^{-1}\left[\frac{2{\left(25-4m\right)}^{\frac{1}{2}}}{\left|4+m\right|}\right]$

Hence, this is the answer.