A homogeneous equation of the second degree is an equation where every term has a total degree of 2.
ax² + 2hxy + by² = 0
This equation always represents a pair of straight lines passing through the origin (0, 0).
We can see this by dividing by x² (assuming x ≠ 0):
a + 2h(y/x) + b(y/x)² = 0
This is a quadratic equation in (y/x). Let m = y/x (the slope).
bm² + 2hm + a = 0
If the roots of this are m₁ and m₂, then:
m₁ + m₂ = -2h/b
m₁m₂ = a/b
The two lines are y/x = m₁ and y/x = m₂, or y = m₁x and y = m₂x.
If θ is the angle between the two lines y = m₁x and y = m₂x, we know from Unit 2 that tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)|.
We can write (m₁ - m₂)² = (m₁ + m₂)² - 4m₁m₂
(m₁ - m₂)² = (-2h/b)² - 4(a/b) = (4h² - 4ab) / b²
|m₁ - m₂| = sqrt(4h² - 4ab) / |b| = 2*sqrt(h² - ab) / |b|
And 1 + m₁m₂ = 1 + a/b = (a + b) / b
Substituting these in:
tan(θ) = | 2 * sqrt(h² - ab) / (a + b) |
This formula gives the angle between the lines represented by ax² + 2hxy + by² = 0.
The lines are parallel or coincident if the angle θ = 0, which means tan(θ) = 0. This happens when the numerator is zero.
Condition: h² - ab = 0 or h² = ab
This means the quadratic for m has equal roots, so m₁ = m₂.
The lines are perpendicular if the angle θ = 90°, which means tan(θ) is undefined. This happens when the denominator is zero.
Condition: a + b = 0
(i.e., Coefficient of x² + Coefficient of y² = 0)
This corresponds to m₁m₂ = a/b = -b/b = -1.
The combined equation of the two lines (which are also perpendicular to each other) that bisect the angles between ax² + 2hxy + by² = 0 is given by a standard formula:
Equation of Angle Bisectors: (x² - y²) / (a - b) = xy / h
y - m₁x = 0 and y - m₂x = 0.| (y - m₁x) / sqrt(1 + m₁²) | = | (y - m₂x) / sqrt(1 + m₂²) |(y - m₁x) / sqrt(1 + m₁²) = + (y - m₂x) / sqrt(1 + m₂²)
(y - m₁x) / sqrt(1 + m₁²) = - (y - m₂x) / sqrt(1 + m₂²)m₁ + m₂ = -2h/b and m₁m₂ = a/b eventually leads to the formula (x² - y²) / (a - b) = xy / h.The most general form of a second-degree equation is:
ax² + 2hxy + by² + 2gx + 2fy + c = 0
This equation can represent a circle, a parabola, an ellipse, a hyperbola, or, in a "degenerate" case, a pair of straight lines.
This general equation represents a pair of straight lines if a specific condition is met. This condition is the most important part of this unit.
The condition can be expressed in two ways:
1. The "long" formula:
abc + 2fgh - af² - bg² - ch² = 0
2. The determinant form (Δ):
The determinant Δ (delta) must be zero.
Δ == 0
a h g h b f g f c
Note: If this condition is met, the angle θ between the two lines is still given by the same formula as the homogeneous case:
tan(θ) = | 2 * sqrt(h² - ab) / (a + b) |
This is because the first-degree terms (2gx + 2fy + c) only *translate* the lines away from the origin, they do not *rotate* them, so the angle between them remains the same.
ax² + 2hxy + by² = 0. Always two lines through the origin.ax² + 2hxy + ... + c = 0. Represents lines *only if* the big determinant Δ = 0.tan(θ) = | 2*sqrt(h² - ab) / (a + b) |. Works for both homogeneous and general equations.a + b = 0. (Sum of x² and y² coefficients is 0).h² = ab.(x² - y²) / (a - b) = xy / h. This is a must-know formula.abc + 2fgh - af² - bg² - ch² = 0. You must memorize this to test if the general equation represents lines.