Foot, Image ,Distance and Angle
Angle between two lines
(i) If \(\theta\) is an acute angle between the two non- vertical lines having slopes \(m_1\) and \(m_2\) then tan\(\theta\) =\(|\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}|\)
(ii) If \(\theta\) is an acute angle between the lines \({a_1}x + {b_1}y + {c_1}\)=0 and \({a_2}x + {b_2}y + {c_2}\) =0,then
\(cos\theta=|\frac{{{a_1}{a_2} + {b_1}{b_2}}}{{\sqrt {{a_1}^2 + {b_1}^2} .\sqrt {{a_2}^2 + {b_2}^2} }}|\) and
tan \(\theta\) = \(|\frac{{{a_1}{b_2} - {a_2}{b_1}}}{{{a_1}{a_2} + {b_1}{b_2}}}|\),other angle between the lines
(iii)The slope m of a line which is equally inclined with two intersecting lines of slopes \(m_1\) and \(m_2\) is given by
\(\frac{{{m_1} - m}}{{1 + m{m_1}}} = \frac{{m - {m_2}}}{{1 + m{m_2}}}\)
(iv)Consider two lines \({L_1} = {a_1}x + {b_1}y + {c_1}\)=0 and \({L_2} = {a_2}x + {b_2}y + {c_2}\)=0
a) Lines are parallel iff \(\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}}\)
b)lines are coincident iff \(\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}\)
c) Lines are perpendicular iff \({a_1}{a_2} + {b_1}{b_2} = 0\)
d) Lines are equally inclined with x-axis iff \(\frac{{{a_1}}}{{{a_2}}} = \frac{{ - {b_1}}}{{{b_2}}}\)