Conic Section Classification
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Direct answer
Eccentricity is the single number that separates the conic sections: the focus-directrix definition SP = e × PM gives a circle at e = 0, an ellipse for 0 < e < 1, a parabola at e = 1 and a hyperbola for e > 1, with e = √2 marking the rectangular hyperbola. The standard forms: y^2 = 4ax (parabola), x^2/a^2 + y^2/b^2 = 1 with b^2 = a^2(1 − e^2) (ellipse), x^2/a^2 − y^2/b^2 = 1 with b^2 = a^2(e^2 − 1) (hyperbola). From ax^2 + 2hxy + by^2 + ..., classify by the discriminant invariant: h^2 < ab an ellipse, h^2 = ab a parabola, h^2 > ab a hyperbola, a + b = 0 rectangular.
What you must remember
- Definition: SP = e × PM, with S the focus, P a point on the curve and M the foot of the perpendicular from P to the directrix — every standard equation descends from it.
- Parabola: y^2 = 4ax has focus (a, 0), directrix x = −a and latus rectum 4a; eccentricity exactly 1.
- Ellipse: x^2/a^2 + y^2/b^2 = 1 with a > b carries foci (±ae, 0), b^2 = a^2(1 − e^2), and focal distances summing to 2a.
- Hyperbola: x^2/a^2 − y^2/b^2 = 1 carries foci (±ae, 0), b^2 = a^2(e^2 − 1), and focal distances differing by 2a.
- Rectangular hyperbola: a = b forces e = √2; xy = c^2 is the same creature with the coordinate axes as asymptotes.
- Discriminant test: on ax^2 + 2hxy + by^2 + ..., compare h^2 with ab: less than for ellipse-type (circle when a = b, h = 0), equal for parabola-type, greater for hyperbola-type; a + b = 0 certifies rectangular.
- Degenerates: an ellipse can collapse to a point and a "hyperbola" can be a line pair — the full equation, not just its quadratic part, decides.
How the chapter is examined
JEE Main tests direct classification and parameter read-offs: identify the conic, quote eccentricity, foci, directrix. JEE Advanced rotates the frame — literally: equations carrying an xy term are classified through h^2 versus ab, sometimes after a rotation whose angle satisfies tan 2α = 2h/(a − b). The recurring losses: calling the larger denominator b in an ellipse (a is always the semi-major axis, the larger one); quoting e = √2 for a hyperbola that is not rectangular; and overlooking degenerate cases, where the discriminant says "ellipse" but the equation has no real points. The identity xy = c^2 = a rectangular hyperbola with e = √2 is the single most quotable line in the chapter — several JEE Main sessions have asked nothing more.
Frequently asked questions
What eccentricity defines each conic type?
Circle e = 0, ellipse 0 < e < 1, parabola e = 1, hyperbola e > 1, rectangular hyperbola e = √2.
What is the eccentricity of x^2/16 + y^2/9 = 1?
√7/4, from e = √(1 − b^2/a^2) with a = 4, b = 3.
How does the discriminant test classify ax^2 + 2hxy + by^2?
Compare h^2 with ab: smaller means ellipse-type, equal means parabola-type, larger means hyperbola-type.
What is the eccentricity of a rectangular hyperbola?
Exactly √2, equivalent to a = b or to perpendicular asymptotes.
What curve is xy = 4?
A rectangular hyperbola referred to its asymptotes — the coordinate axes serve as them.