Conic Sections

🎯 Key Points

• Eccentricity sorts all conics on one scale: e=0 (circle) → 0<e<1 (ellipse) → e=1 (parabola) → e>1 (hyperbola)
• Ellipse: c²=a²-b² (foci INSIDE the curve, sum of focal distances=2a constant); Hyperbola: c²=a²+b² (foci OUTSIDE relative to vertices, DIFFERENCE of focal distances=2a constant) - opposite sign in the c² relation is the key distinguishing feature
• Parabola y²=4ax: focus (a,0), directrix x=-a, latus rectum length 4a - latus rectum is the focal chord PERPENDICULAR to the axis
• For an ellipse x²/a²+y²/b²=1, the LARGER denominator tells you which axis is the major axis - don't assume x² always goes with the major axis

Conic Sections

A conic section is the curve obtained when a plane intersects a double-napped right circular cone. Depending on the angle of the cutting plane relative to the axis of the cone, the resulting curve is a circle, parabola, ellipse, or hyperbola. These curves can also be defined as the locus of a point whose distance from a fixed point (focus) bears a constant ratio (eccentricity) to its distance from a fixed line (directrix).

Circle (Quick Recap)

• A circle is the set of points equidistant from a fixed point (center).
• Standard equation with center (h, k) and radius r: (x - h)² + (y - k)² = r²
• Center at origin: x² + y² = r²
• General form: x² + y² + 2gx + 2fy + c = 0, with center (-g, -f) and radius √(g² + f² - c)
• A circle is a special conic with eccentricity e = 0.

Parabola

• A parabola is the locus of a point that moves so that its distance from a fixed point (focus) equals its distance from a fixed line (directrix). Eccentricity e = 1.
• Standard equation: y² = 4ax (opens rightward, a > 0)
• Focus: (a, 0)   Directrix: x = -a   Vertex: (0, 0)   Axis: x-axis
• Latus rectum (the focal chord perpendicular to the axis) has length 4a, with endpoints (a, 2a) and (a, -2a).
• Parametric form: x = at², y = 2at
• Other orientations: y² = -4ax (opens left), x² = 4ay (opens up), x² = -4ay (opens down), each with focus and directrix adjusted accordingly.

Ellipse

• An ellipse is the locus of a point such that the sum of its distances from two fixed points (foci) is constant. Eccentricity 0 < e < 1.
• Standard equation: x²/a² + y²/b² = 1, with a > b > 0 (major axis along x-axis)
• Relationship between a, b, c: c² = a² - b², where c is the distance from center to each focus.
• Foci: (±c, 0)   Vertices: (±a, 0)   Eccentricity: e = c/a
• Directrices: x = ±a/e
• Latus rectum length: 2b²/a
• Length of major axis = 2a, length of minor axis = 2b
• When the major axis is along the y-axis (b > a in the equation form x²/b² + y²/a² = 1 with a > b), foci are (0, ±c) and the roles of axes swap.
• Sum of focal distances from any point on the ellipse to the two foci = 2a (constant).

Hyperbola

• A hyperbola is the locus of a point such that the absolute difference of its distances from two fixed points (foci) is constant. Eccentricity e > 1.
• Standard equation: x²/a² - y²/b² = 1 (transverse axis along x-axis)
• Relationship between a, b, c: c² = a² + b², where c is the distance from center to each focus.
• Foci: (±c, 0)   Vertices: (±a, 0)   Eccentricity: e = c/a (always > 1)
• Latus rectum length: 2b²/a
• Asymptotes: y = ±(b/a)x, lines that the hyperbola approaches but never touches as it extends to infinity
• Difference of focal distances from any point on the hyperbola to the two foci = 2a (constant, in absolute value).

Comparison of Conics by Eccentricity

Key Tips for Problem Solving

• To identify a conic from its general equation, compare coefficients with the standard forms after shifting/rotating if needed.
• Always check whether the major axis is along the x-axis or y-axis by comparing the denominators under x² and y² in the ellipse equation (larger denominator indicates the axis direction).
• For a hyperbola, b is not necessarily less than a; only the relation c² = a² + b² matters since c > a always.

🚀 JEE Advanced Edge

Why the focal-distance-sum/difference properties are the "real" definitions: The focus-directrix definition (distance to focus = e × distance to directrix) and the focal-distance-sum/difference definition are EQUIVALENT for each conic, but the sum/difference form (2a constant) is usually faster for proving locus problems, since it avoids setting up a directrix line at all - recognizing which definition to invoke saves significant algebra in JEE-level locus questions.

Rectangular hyperbola as a special case: When a=b in x²/a²-y²/b²=1, the hyperbola becomes "rectangular" (asymptotes y=±x are perpendicular), and its eccentricity is always e=√2 regardless of the value of a - a useful shortcut when a problem states "rectangular hyperbola" without giving explicit asymptote directions.

Worked problem: Find the eccentricity of the ellipse 4x²+9y²=36. Approach: Divide by 36: x²/9+y²/4=1, so a²=9, b²=4 (a²>b², major axis along x). c²=a²-b²=9-4=5, so c=√5. Eccentricity e=c/a=√5/3.

Worked Example: Parabola - Focus, Directrix, Length of Latus Rectum

For the parabola y² = 12x, find the focus, equation of the directrix, and length of the latus rectum.

Compare with y² = 4ax: 4a = 12 → a = 3. Focus = (a, 0) = (3, 0). Directrix: x = −a → x = −3. Length of latus rectum = 4a = 12.

Worked Example: Equation of a Circle

Find the equation of the circle passing through (1, 0), (−1, 0), and (0, 1).

General form: x² + y² + Dx + Ey + F = 0. Substituting (1,0): 1 + D + F = 0. (−1,0): 1 − D + F = 0. Subtracting: 2D = 0 → D = 0, F = −1. (0,1): 1 + E − 1 = 0 → E = 0. Equation: x² + y² = 1 (unit circle centred at origin). Three points determine a unique circle.

Sections of a Cone and Degenerate Conics

• The conic sections arise by cutting a double-napped right circular cone with a plane; the tilt of the plane relative to the axis fixes the curve produced.
• A plane perpendicular to the axis gives a circle; a slight tilt gives an ellipse.
• A plane parallel to a slant generator (side) of the cone gives a parabola; a steeper plane cutting both nappes gives a hyperbola.
• When the cutting plane passes through the vertex, the section degenerates to a point, a single straight line, or a pair of intersecting straight lines.
• These degenerate cases correspond to the same equations with special parameter values, unifying them with the standard conics.

Eccentricity: The Unifying Parameter

• Eccentricity e is the constant ratio of a point's distance from the focus to its distance from the directrix, and it classifies every conic.
• Circle: e = 0; ellipse: 0 less than e less than 1; parabola: e = 1; hyperbola: e greater than 1.
• For an ellipse and a hyperbola, e = c/a, where c is the focus distance from the centre and a is the semi-major (or semi-transverse) length.
• A larger eccentricity means a more elongated or open curve; e near 0 gives an almost circular shape.
• Eccentricity, together with a focus and a directrix, provides a single locus definition valid for all conics.

Latus Rectum of the Conics

• The latus rectum is the chord through a focus drawn perpendicular to the major (or transverse) axis; its length gauges how wide the conic opens at the focus.
• Parabola y² = 4ax: length of latus rectum = 4a, with endpoints (a, 2a) and (a, -2a).
• Ellipse x²/a² + y²/b² = 1: length of each latus rectum = 2b²/a.
• Hyperbola x²/a² - y²/b² = 1: length of each latus rectum = 2b²/a as well.
• The ellipse and hyperbola each have two latus recta, one at each focus, while the parabola has one.

Standard Forms of the Circle

• A circle with centre (h, k) and radius r has equation (x - h)² + (y - k)² = r²; centred at the origin it becomes x² + y² = r².
• The general equation x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius sqrt(g² + f² - c).
• It is a real circle only when g² + f² - c is greater than 0; equal to 0 gives a point circle, and less than 0 gives no real locus.
• A circle passing through the origin has c = 0, since substituting (0, 0) must satisfy the equation.
• A circle is the special conic of eccentricity e = 0, the limiting case of an ellipse whose two foci coincide at the centre.