Conic Sections - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Find Equation of Conic from Given Conditions

Pattern

Given focus, directrix, eccentricity, or other geometric data, find the equation of the conic

How to recognize

Problem gives partial information about a conic (focus, vertex, eccentricity, endpoints of axes) and asks for the full equation

→ Identify conic type from eccentricity (e < 1: ellipse, e = 1: parabola, e > 1: hyperbola). Use standard form and given conditions to find a, b, or orientation.

Tangent and Normal Equations

Pattern

Find the equation of tangent or normal to a conic at a given point or with a given slope

How to recognize

Asks for tangent/normal line to parabola, ellipse, or hyperbola. May give point on curve, external point, or slope.

→ Use slope form (y = mx + a/m for parabola) or point form (T = 0). For external point, solve for m using tangency condition. Verify with discriminant = 0.

Locus Problems on Conics

Pattern

Find the locus of a point satisfying a geometric condition involving a conic

How to recognize

A point moves under a constraint related to a conic (midpoint of chord, foot of perpendicular, intersection of normals), find its path

→ Let the moving point be (h, k). Express the constraint using parametric or Cartesian form. Eliminate parameters to get the relation in h and k, then replace h with x and k with y.

Chord Properties and Chord of Contact

Pattern

Find mid-point of chord, chord of contact from external point, or chord with given mid-point

How to recognize

Involves chord of a conic with conditions on mid-point, or chord of contact from an external point, or chord bisected at a given point

→ For chord of contact: use T = 0 from external point. For chord with given mid-point: use T = S₁. For pair of tangents: SS₁ = T².

Focal Chord Problems

Pattern

Properties of a chord passing through the focus of a conic

How to recognize

Chord passes through focus, or involves endpoints of a focal chord, latus rectum, or harmonic mean of focal distances

→ For parabola: use t₁t₂ = -1. Apply 1/SP + 1/SQ = 2/l. Length of focal chord = a(t - 1/t)². For ellipse/hyperbola: use focal distance formulas SP = a + ex₁.

Eccentricity and Related Problems

Pattern

Find eccentricity or use eccentricity conditions to determine the conic

How to recognize

Asks for eccentricity, or gives a relationship between a, b, e and asks to determine the conic or a parameter

→ Ellipse: e = sqrt(1 - b²/a²), range (0,1). Hyperbola: e = sqrt(1 + b²/a²), range (1, infinity). Use b² = a²(1 - e²) for ellipse, b² = a²(e² - 1) for hyperbola.

Parametric Form Applications

Pattern

Use parametric representation to solve tangent, normal, chord, or area problems on conics

How to recognize

Problem involves parametric points on the conic, or asks to prove a property using parametric coordinates

→ Substitute parametric coordinates. For parabola: tangent at t is ty = x + at². For ellipse: tangent at theta is (x cos theta)/a + (y sin theta)/b = 1.

Director Circle and Auxiliary Circle

Pattern

Find the locus of perpendicular tangents or use auxiliary circle properties

How to recognize

Asks for locus where two tangents are perpendicular, or involves the auxiliary circle x² + y² = a²

→ Director circle of ellipse: x² + y² = a² + b². Director circle of hyperbola: x² + y² = a² - b² (exists only if a > b). Auxiliary circle: x² + y² = a².

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