Pair of Straight Lines
5.0 Angle between the lines represented by general equation
5.0 Angle between the lines represented by general equation
The angle $\theta $ between the two lines representing by a general equation is the same as that between the two lines represented by its homogeneous part only i.e., $$\theta = {\tan ^{ - 1}}\{ \frac{{2\sqrt {({h^2} - ab)} }}{{\left| {a + b} \right|}}\} $$
Proof: Let $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$ be the lines represented by $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0...(1)$$ where ${m_1} = \tan \alpha $ and ${m_2} = \tan \beta $. Then, we can write $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c \equiv (y - {m_1}x - {c_1})(y - {m_2}x - {c_2})$$
On comparing the coefficients we get, $${m_1} + {m_2} = - \frac{{2h}}{b}...(1)$$ and $${m_1} \times {m_2} = \frac{a}{b}...(2)$$
From equation $(1)$ and $(2)$,
$$\left| {{m_1} - {m_2}} \right| = \sqrt {\{ {{({m_1} + {m_2})}^2} - 4{m_1}{m_2}\} } = \frac{2}{b}\sqrt {({h^2} - ab)}...(3) $$
As we know that the angle between two lines is calculated using the formulae $$\tan \theta = \left| {\frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}}} \right|$$
Put the values from equation $(2)$ and $(3)$, we get $$\begin{equation} \begin{aligned} \tan \theta = \left| {\frac{{\frac{2}{b}\sqrt {({h^2} - ab)} }}{{1 + \frac{a}{b}}}} \right| = \frac{{2\sqrt {({h^2} - ab)} }}{{a + b}} \\ \theta = {\tan ^{ - 1}}\{ \frac{{2\sqrt {({h^2} - ab)} }}{{\left| {a + b} \right|}}\} \\\end{aligned} \end{equation} $$
Conditions:
- The lines are perpendicular if $\theta = \frac{\pi }{2}$ i.e., $\cot \theta = 0$ $$\begin{equation} \begin{aligned} \frac{{\left| {a + b} \right|}}{{2\sqrt {({h^2} - ab)} }} = 0 \\ a + b = 0 \\ {\text{coefficient of }}{x^2} + {\text{coefficient of }}{y^2} = 0 \\\end{aligned} \end{equation} $$
- The lines are parallel, if $\theta = {0^ \circ }{\text{ or }}\pi $ i.e., $\tan \theta = 0$$$\begin{equation} \begin{aligned} \frac{{2\sqrt {({h^2} - ab)} }}{{\left| {a + b} \right|}} = 0 \\ {h^2} - ab = 0 \\ {h^2} = ab \\\end{aligned} \end{equation} $$
- The lines are coincident if $${h^2} - ab = 0,{\text{ }}{g^2} - ac = 0{\text{ and }}{f^2} - bc = 0$$
