🎯 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
All four conics form a single family distinguished only by eccentricity: a circle is the most "closed" (e=0), an ellipse is an elongated closed curve, a parabola is the borderline open curve (e=1), and a hyperbola has two separate open branches (e>1).
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
| Conic | Eccentricity (e) | Shape |
|---|---|---|
| Circle | e = 0 | Perfectly round, both foci coincide at center |
| Ellipse | 0 < e < 1 | Oval, two distinct foci inside the curve |
| Parabola | e = 1 | Open curve, one focus and one directrix |
| Hyperbola | e > 1 | Two separate open branches, asymptotic lines |
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.