Introduction to Pair of Lines
In coordinate geometry, the equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin. These lines are inclined at certain angles to each other. The angle bisectors are the lines that divide these angles into two equal parts.
The Derivation
To find the combined equation of the bisectors of the angles between the lines given by $ax^2 + 2hxy + by^2 = 0$, we follow these steps:
1. Identify individual lines
Let the lines be $y - m_1x = 0$ and $y - m_2x = 0$. Then, $(y - m_1x)(y - m_2x) = y^2 - (m_1+m_2)xy + m_1m_2x^2 = 0$. Comparing this with $ax^2 + 2hxy + by^2 = 0$ (or $\frac{a}{b}x^2 + \frac{2h}{b}xy + y^2 = 0$):
- $m_1 + m_2 = -\frac{2h}{b}$
- $m_1m_2 = \frac{a}{b}$
2. Equation of an Angle Bisector
For any point $(x, y)$ on a bisector, the perpendicular distance to the lines must be equal: $\left|\frac{y - m_1x}{\sqrt{1 + m_1^2}}\right| = \left|\frac{y - m_2x}{\sqrt{1 + m_2^2}}\right|$
Squaring both sides: $(y - m_1x)^2 (1 + m_2^2) = (y - m_2x)^2 (1 + m_1^2)$
3. Simplifying the Equation
Expanding both sides leads to: $(y^2 - 2m_1xy + m_1^2x^2)(1 + m_2^2) = (y^2 - 2m_2xy + m_2^2x^2)(1 + m_1^2)$
After rearranging and grouping terms with $x^2, xy,$ and $y^2$, we use the identity $(m_1 - m_2)^2 = (m_1 + m_2)^2 - 4m_1m_2$. Substituting the values of $m_1+m_2$ and $m_1m_2$ derived in step 1, we arrive at the standard result.
The Final Formula
The combined equation of the angle bisectors of the pair of lines $ax^2 + 2hxy + by^2 = 0$ is:
$\frac{x^2 - y^2}{a - b} = \frac{xy}{h}$
or more commonly written as:
$\frac{x^2 - y^2}{a - b} = \frac{xy}{h} \implies h(x^2 - y^2) = (a - b)xy$
Intuition
- Symmetry: The bisectors are always perpendicular to each other because the lines they bisect have a specific angular relationship.
- Homogeneity: Since the original equation is homogeneous (all terms are degree 2), the bisectors must also form a homogeneous equation of degree 2, passing through the origin.
- Coefficients: If $a = b$, the equation simplifies to $xy = 0$, meaning the bisectors are the coordinate axes themselves (the lines $x=0$ and $y=0$).