Pair of Straight Lines

Take the equation x^2 - 5xy + 6y^2 = 0. This is a second-degree equation in x and y — it has x^2, xy, and y^2 terms, and it looks like it should describe some kind of curve. Conics — ellipses, parabolas, hyperbolas — are second-degree curves. But this one is different.

Factor the left side. Treat the equation as a quadratic in x, with y as a constant:

A product is zero when at least one factor is zero. So the equation is satisfied when x - 2y = 0 or x - 3y = 0. These are two straight lines through the origin — y = x/2 and y = x/3.

A single equation, of degree two, has quietly been describing two lines all along. Not a curve — two lines. The second-degree equation is a disguised pair of first-degree equations multiplied together.

This is not a coincidence. There is a systematic theory that tells you exactly when a second-degree equation represents a pair of lines, what the angle between the lines is, and what the bisectors are. Indian textbooks cover this as a separate chapter because it appears frequently in JEE and board exams — and because the underlying algebra is clean and beautiful.

Homogeneous equations: lines through the origin

Why homogeneous?

An equation is homogeneous of degree n if every term has total degree exactly n. The equation ax^2 + 2hxy + by^2 = 0 is homogeneous of degree 2: every term has degree 2 (counting both x and y together). There are no linear terms (x, y) and no constant term.

The key property of a homogeneous equation is: if the point (x_0, y_0) satisfies it, then so does (tx_0, ty_0) for any scalar t.

So if one point is on the curve, the entire line from the origin through that point is on the curve. The graph of a homogeneous equation is a union of lines through the origin.

For degree 2, there can be at most two such lines — and the conditions for exactly two, exactly one (a repeated line), or none are easy to derive.

Factoring the homogeneous equation

Take ax^2 + 2hxy + by^2 = 0. If b \neq 0, divide through by y^2 (allowed when y \neq 0) to get a quadratic in x/y:

Set m = x/y. This is a standard quadratic am^2 + 2hm + b = 0, and its solutions are

Each solution m = m_i gives a line x = m_i y, or equivalently x - m_i y = 0. If m_1 and m_2 are the two roots, the original equation factors as

and the two lines are x - m_1 y = 0 and x - m_2 y = 0.

The discriminant condition

The quadratic am^2 + 2hm + b = 0 has:

• Two distinct real roots when h^2 - ab > 0: two distinct lines through the origin.
• One repeated root when h^2 - ab = 0: a single line, counted twice (a "repeated" or "coincident" pair).
• No real roots when h^2 - ab < 0: the equation represents only the origin (0, 0) — no real lines.

Useful relations from the roots

By Vieta's formulas for the quadratic am^2 + 2hm + b = 0, the two slopes m_1, m_2 satisfy

These are enormously useful. They let you compute the angle between the lines, the condition for perpendicularity, and the bisector equations — all without explicitly finding m_1 and m_2.

Angle between the pair of lines

The angle \theta between two lines with slopes m_1 and m_2 satisfies

To express this in terms of a, h, b, notice that

So |m_1 - m_2| = \frac{2\sqrt{h^2 - ab}}{|a|}. Also, 1 + m_1 m_2 = 1 + b/a = (a + b)/a.

Therefore:

Special case: perpendicular lines. The lines are perpendicular when \theta = 90°, i.e., \tan\theta is undefined. This happens when 1 + m_1 m_2 = 0, which means a + b = 0. So:

The pair ax^2 + 2hxy + by^2 = 0 consists of perpendicular lines if and only if a + b = 0 — the coefficients of x^2 and y^2 sum to zero.

This is a beautiful and easily remembered condition. For example, x^2 - y^2 = 0 gives a = 1, b = -1, and a + b = 0. The lines are x - y = 0 and x + y = 0, which are indeed perpendicular (slopes +1 and -1).

Bisectors of the pair

Given the pair ax^2 + 2hxy + by^2 = 0, the bisectors of the angles between the two lines have a surprisingly elegant equation.

Deriving the bisector equation

Let the two lines be y = m_1 x and y = m_2 x, i.e., m_1 x - y = 0 and m_2 x - y = 0. The angle bisectors satisfy the equal-distance condition:

Squaring both sides (to eliminate the \pm and the square roots simultaneously):

Expand and rearrange. After careful algebra (expanding the squares, cross-multiplying, and using m_1 + m_2 = -2h/a and m_1 m_2 = b/a), the result simplifies to:

Why this is elegant. The bisector equation is itself a homogeneous second-degree equation — a pair of lines through the origin. The two bisectors are perpendicular (check: in hx^2 - (a - b)xy - hy^2 = 0, the coefficient of x^2 is h and the coefficient of y^2 is -h, and h + (-h) = 0, confirming perpendicularity).

Important note. This formula requires a \neq b and h \neq 0. If a = b, the bisectors are xy = 0, i.e., the coordinate axes. If h = 0, the original lines are x^2 - (b/a)y^2 = 0, i.e., symmetric about the axes, and the bisectors are x^2 - y^2 = 0.

Worked examples

The general second-degree equation

When does it represent a pair of lines?

The general second-degree equation in x and y is

This represents a pair of straight lines (not necessarily through the origin) if and only if the following determinant is zero:

Why this condition? If the equation represents two lines (\ell_1 x + m_1 y + n_1)(\ell_2 x + m_2 y + n_2) = 0, expanding the product and matching coefficients with the general form gives six equations in six unknowns (\ell_1, m_1, n_1, \ell_2, m_2, n_2). Eliminating these unknowns systematically produces the determinant condition. The determinant is called the discriminant of the second-degree equation (not to be confused with the discriminant of a quadratic in one variable).

Extracting the individual lines

Once you know \Delta = 0 and the equation represents a pair of lines, you can find the individual lines. Two approaches:

Method 1: Factor directly. Treat the equation as a quadratic in x (grouping y-terms as constants):

Use the quadratic formula in x. The discriminant (in x) must be a perfect square in y — this is guaranteed by \Delta = 0.

Method 2: Find the intersection point first. Partially differentiate the equation with respect to x and y:

Solve this 2 \times 2 system to find the point of intersection (\alpha, \beta) of the two lines. Then shift the origin to (\alpha, \beta) — the equation becomes homogeneous, and you can factor it using the methods above.

The homogeneous part determines the angle

An important observation: the angle between the two lines in ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 depends only on a, h, b — the homogeneous second-degree part. The linear and constant terms shift the lines but do not rotate them.

So the angle formula is exactly the same as for the homogeneous case:

And the perpendicularity condition is still a + b = 0.

Common confusions

• "Every second-degree equation represents a pair of lines." Far from it. A second-degree equation is generically a conic (ellipse, parabola, or hyperbola). A pair of lines is the degenerate case — the conic collapses into two lines. This happens only when \Delta = 0.

• "ax^2 + 2hxy + by^2 = 0 always gives two lines." Only when h^2 \geq ab. If h^2 < ab, the only real solution is x = y = 0 — just the origin, no lines. For instance, x^2 + y^2 = 0 has only the solution (0, 0) over the reals.

• "The pair-of-lines angle formula uses a - b." No, the denominator is |a + b|. The bisector equation uses a - b. Mixing these up is the single most common error in this chapter.

• "The bisectors of a pair of lines depend on all six coefficients." For lines through the origin (ax^2 + 2hxy + by^2 = 0), the bisector equation h(x^2 - y^2) = (a - b)xy uses only a, h, b. For the general equation, you first shift to the intersection point, then apply the same formula to the homogeneous part.

• "If \Delta = 0, the two lines are distinct." Not necessarily. You also need h^2 - ab > 0 for the lines to be distinct. If h^2 = ab and \Delta = 0, the "pair" is really one line counted twice (coincident lines). If h^2 < ab and \Delta = 0, the equation represents a single point.

Going deeper

The homogenisation technique

Here is a powerful application of the pair-of-lines theory. Suppose you have a second-degree curve S: ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 and a line L: \ell x + my + n = 0. The line meets the curve at two points. The pair of lines joining the origin to these two intersection points can be found by homogenising the curve equation using the line.

From the line equation, 1 = -(\ell x + my)/n (assuming n \neq 0). Now replace every occurrence of the "constant" in the curve equation using this expression. Specifically, replace 2gx with 2gx \cdot 1 = 2gx \cdot \frac{-(\ell x + my)}{n}, replace 2fy similarly, and replace c with c \cdot 1^2 = c \cdot \frac{(\ell x + my)^2}{n^2}. The result is a homogeneous second-degree equation in x and y — it represents the pair of lines OP and OQ, where P and Q are the intersection points.

This technique is especially useful in circle problems: "find the angle subtended at the origin by a chord of a given circle" reduces to homogenising the circle equation with the chord equation, then applying the pair-of-lines angle formula.

Connection to conics

A conic section is described by the general second-degree equation ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. The nature of the conic is determined by the discriminant \Delta and the quantity h^2 - ab:

A pair of lines is therefore a degenerate conic — the conic that has collapsed. The determinant \Delta = 0 is the degeneracy condition. In the language of Bhaskara II's Bijaganita, which systematically classified second-degree equations, these degenerate cases were already recognised as special situations where the "curve" loses its curvature.

Why the determinant 3 \times 3?

The matrix

is called the matrix of the conic. It encodes the entire second-degree equation: the equation can be written as \mathbf{x}^T M \mathbf{x} = 0 where \mathbf{x} = (x, y, 1)^T (using homogeneous coordinates). The condition \det M = 0 means the matrix is singular — it has a null vector. That null vector, when interpreted geometrically, is the intersection point of the two lines (in homogeneous coordinates). A singular conic matrix corresponds to a degenerate conic, which is precisely a pair of lines (or a repeated line, or a point).

This is a deep connection between linear algebra and geometry: the determinant of a 3 \times 3 symmetric matrix deciding whether a curve is really a pair of lines.

Where this leads next

The pair-of-lines theory is a gateway to several important topics:

• Angle Bisectors — the bisector formula h(x^2 - y^2) = (a - b)xy is a special case of the general bisector theory for two lines.
• Angle and Conditions — the angle between two lines, whether given individually or as a pair, uses the same underlying formula.
• Quadratic in Two Variables — the general second-degree equation is the starting point for all conic sections.
• Determinants — the \Delta = 0 condition is one of the most natural applications of 3 \times 3 determinants in coordinate geometry.
• Circle - Standard Forms — homogenisation of a circle equation with a line produces a pair of lines, connecting this chapter directly to circle geometry.