Pair of Straight Lines

In earlier years, you studied how to write the equation of a single straight line — y = mx + c, ax + by + c = 0, and so on. But what happens when two lines overlap? Or pass through the same point? Can we represent both of them with a single equation? Yes — and that's exactly what this chapter is about. A pair of straight lines is essentially two distinct lines that pass through a common point, combined into a single second-degree equation. The cool thing is that you can look at a quadratic equation in x and y and recognise whether it represents two straight lines, and if so, find their individual equations. This chapter starts with how to form the combined equation when you know the two individual lines. If L₁ = 0 and L₂ = 0 are two lines, then their product L₁ × L₂ = 0 represents both lines together. When we multiply these, we get a second-degree homogeneous equation of the form ax² + 2hxy + by² = 0 — provided both lines pass through the origin. This is the homogeneous form, and it's the simplest case. The key parameters are a, h, and b, and from these three numbers we can determine everything: whether the lines are real or imaginary, the angle between them, whether they're parallel or perpendicular, and the individual line equations. The angle between the two lines is given by the formula tan θ = |2√(h² − ab)| / |a + b|. If h² = ab, the two lines are parallel (or coincident). If a + b = 0, the lines are perpendicular. If h² < ab, no real lines exist — the pair is imaginary. So the discriminant h² − ab tells you the nature of the pair. Now, what about lines that DON'T pass through the origin? That's the general second-degree equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0. This represents a pair of straight lines if and only if the determinant condition holds: Δ = abc + 2fgh − af² − bg² − ch² = 0. This is the 'pair of lines condition.' When it holds, you can shift the origin to the point of intersection of the two lines to get back to the homogeneous form. Finding the point of intersection itself involves solving ∂/∂x = 0 and ∂/∂y = 0 of the general equation — which gives the coordinates of the centre. You'll also learn how to find the separate equations of the two lines from the combined equation. The method: treat ax² + 2hxy + by² = 0 as a quadratic in x (or y), solve for x in terms of y using the quadratic formula, and factor. Each factor gives one line. Alternatively, if the general equation is given and the pair condition holds, you can factor the quadratic part, then adjust the linear terms to match the full expression. There's also a method using the formula for the slopes: the slopes m₁ and m₂ satisfy m₁ + m₂ = −2h/b and m₁m₂ = a/b (when the equation is treated as quadratic in y/x). These relationships come from the coefficients and are useful shortcuts. By the end of this chapter, you'll be able to look at any second-degree equation, quickly check if it represents a pair of lines, find the angle between them, determine their individual equations, and understand their geometric relationship. This is a short but powerful chapter — the concepts are limited but the applications range from coordinate geometry to analytic geometry used in computer vision and graphics.