General Second-Degree Equation & Conics: JEE Main

The general second-degree equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 is the unified form that encompasses all conic sections. JEE Main tests it through "identify the conic" questions, "pair of straight lines" problems, and the conditions for specific degeneracies. Students who know the discriminant conditions can classify a given equation in 20 seconds; those who do not spend precious time trying to complete the square.

The Discriminant and Conic Classification

The matrix of the second-degree equation: Δ = |a h g; h b f; g f c| (3×3 determinant). Also define the quantity D = ab − h² (the 2×2 sub-determinant from the quadratic part). Classification by Δ and D: (1) If Δ = 0: the conic is degenerate (breaks into lines or a point). (2) If Δ ≠ 0 and D > 0: ellipse (or circle if a = b and h = 0). (3) If Δ ≠ 0 and D = 0: parabola. (4) If Δ ≠ 0 and D < 0: hyperbola. (5) If Δ ≠ 0, D > 0, and a = b and h = 0: circle. JEE question: given 5x² − 2xy + 5y² + 4x − 2y + 1 = 0, classify the conic. D = 5×5 − (−1)² = 25 − 1 = 24 > 0, a = b = 5, h = −1 ≠ 0 (h ≠ 0 so not a circle) → ellipse.

Eccentricity from the general equation: for the ellipse/hyperbola, find the ratio of eigenvalues of the quadratic form matrix. JEE Main rarely asks for eccentricity from the general equation — it asks for classification. Take a free conic sections mock. See our conic sections guide.

Pair of Straight Lines: The Degenerate Case

When Δ = 0, the general second-degree equation represents a pair of lines (real or imaginary, coincident or distinct). Condition for ax² + 2hxy + by² + 2gx + 2fy + c = 0 to be a pair of straight lines: Δ = abc + 2fgh − af² − bg² − ch² = 0. For the homogeneous part only (g = f = c = 0): ax² + 2hxy + by² = 0 represents a pair of lines through the origin. Slopes of the pair: m₁ + m₂ = −2h/b and m₁m₂ = a/b (Vieta's formulas applied to the quadratic in m). Angle between the pair: tanθ = 2√(h²−ab)/(a+b). The pair is perpendicular when a + b = 0 (sum of coefficients of x² and y² = 0).

Condition for Circle and Rectangular Hyperbola

The general second-degree equation represents a circle when: h = 0 (no xy term) and a = b (equal coefficients of x² and y²). The standard form is then a(x² + y²) + 2gx + 2fy + c = 0, which after dividing by a gives the standard circle form. Centre: (−g/a, −f/a). Radius: √((g/a)² + (f/a)² − c/a). JEE checks whether a given equation is a circle by verifying h = 0 and a = b — this is a direct 5-second check. Rectangular hyperbola: the general second-degree equation represents a rectangular hyperbola when the coefficient of x² plus the coefficient of y² equals zero (a + b = 0), and it is a genuine hyperbola (D < 0, Δ ≠ 0).

Intersection of Conics and Combined Equations

A pair of lines combined with a circle: S ≡ equation of circle = 0, L ≡ equation of chord = 0. The equation S + λL² = 0 (or S + λL = 0 for a diameter) represents a family of conics through the intersection of S and L. JEE uses this as: "write the equation of a conic passing through the intersection of two given conics." The method: S₁ + λS₂ = 0 (gives all conics through the 4 intersection points of two conics S₁ and S₂). For a specific condition (passes through a point, is a circle, etc.), substitute to find λ. For detailed treatment of each conic type see our parabola guide, ellipse guide, and hyperbola guide.