Pair of straight lines

1. Combined Equation of a Pair of Lines

Joint Equation:
The equation $uv = 0$ represents the combined (or joint) equation of the lines $u = 0$ and $v = 0$.

Intersecting Lines u=0 and v=0

Standard Form:
$ax^2 + 2hxy + by^2 = 0$ represents a pair of lines passing through the origin if $h^2 - ab \geq 0$.
Slopes of the Lines ($m_1, m_2$):
$$m_1 = \frac{-h - \sqrt{h^2 - ab}}{b}, \quad m_2 = \frac{-h + \sqrt{h^2 - ab}}{b}$$
Sum and Product of Slopes:
$$m_1 + m_2 = -\frac{2h}{b}, \quad m_1m_2 = \frac{a}{b}$$
Auxiliary Equation:
$bm^2 + 2hm + a = 0$ (Roots are the slopes $m_1$ and $m_2$).

Angle Between Lines passing through Origin

Acute Angle ($\theta$):
If $\theta$ is the acute angle between the lines $ax^2 + 2hxy + by^2 = 0$, then:

$$\tan \theta = \left|\frac{2\sqrt{h^2 - ab}}{a + b}\right|$$
Condition for Coincident Lines:
$h^2 - ab = 0$ (or $h^2 = ab$).
Condition for Perpendicular Lines:
$a + b = 0$.
Condition for Distinct Lines:
$h^2 - ab > 0$.

Standard Form:
$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
Necessary Conditions to Represent a Pair of Lines:
- 1. $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$
- 2. $h^2 - ab \geq 0$
Determinant Form of the Condition:
$$\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0$$
Point of Intersection:
If the general equation represents intersecting lines ($h^2 - ab > 0$), their point of intersection is:

$$\left( \frac{hf - bg}{ab - h^2}, \frac{gh - af}{ab - h^2} \right)$$

Bisectors of Angles:
The joint equation of the bisectors of the angles between the lines $ax^2 + 2hxy + by^2 = 0$ is:

$$hx^2 - (a-b)xy - hy^2 = 0$$
Perpendicular Pair:
The combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by $ax^2 + 2hxy + by^2 = 0$ is:

$$bx^2 - 2hxy + ay^2 = 0$$
Product of Perpendiculars:
The product of lengths of perpendiculars drawn from $P(x_1, y_1)$ to the lines $ax^2 + 2hxy + by^2 = 0$ is:

$$\left|\frac{ax_1^2 + 2hx_1y_1 + by_1^2}{\sqrt{(a-b)^2 + 4h^2}}\right|$$