Q.Find the combined equation of the straight lines whose separate equations are $x - 2y - 3 = 0$ and $x + y + 5 = 0$.

Pair of Straight Lines

A second-degree equation $ax^2+2hxy+by^2=0$ represents two straight lines through the origin; their slopes $m_1,m_2$ satisfy $m_1+m_2=-\dfrac{2h}{b}$ and

$m_1m_2=\dfrac{a}{b}$. The lines are real and distinct when $h^2>ab$, coincident when

$h^2=ab$, and the angle between them is $\tan\theta=\dfrac{2\sqrt{h^2-ab}}{|a+b|}$;

they are perpendicular iff $a+b=0$ and coincident/parallel-behaving iff $h^2=ab$.

The general second-degree equation

$ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of lines exactly when

$\Delta=abc+2fgh-af^2-bg^2-ch^2=0$. When it does, the lines meet at the point

found from $\dfrac{\partial}{\partial x}=0,\ \dfrac{\partial}{\partial y}=0$, i.e.

$\Big(\dfrac{hf-bg}{ab-h^2},\dfrac{gh-af}{ab-h^2}\Big)$, and they are parallel when

$h^2=ab$ together with $bg^2=af^2$. The pair of angle-bisectors of

$ax^2+2hxy+by^2=0$ is $\dfrac{x^2-y^2}{a-b}=\dfrac{xy}{h}$.

The pair-of-straight-lines equation builds on the NCERT Class 11 Mathematics "Straight Lines" chapter and is an important topic for JEE Main, JEE Advanced and state CETs. "Pair of straight lines formulas" and "angle between pair of lines through origin" are typical searches this concept answers.