Parametric Equations of Conics (JEE Advanced)

“parametric equations of a circle; Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: equations of conic sections (parabola, ellipse and hyperbola) in standard forms.

JEE Main focuses on standard Cartesian equations of conics, but Advanced expects you to switch fluently between Cartesian and parametric forms. Many Advanced problems on tangents, normals, and locus become much simpler when you use parametric coordinates.

• A parameter is just a helper variable. For a circle, the angle θ tells you where you are on the curve. For a parabola, the parameter t controls both x and y.
• Each conic has a standard parametric form. Memorize these forms along with how the parameter relates to the geometry of the curve.
• Tangent and normal equations at a parametric point have clean formulas. These are faster than implicit differentiation in most exam problems.
• The parametric form helps when you need to find the locus of a point defined by conditions on the conic, like the midpoint of a chord or the foot of a perpendicular.
• For ellipse and hyperbola, the parameter θ is NOT the actual angle from the center to the point. It is called the eccentric angle and has a specific geometric meaning.
• When two parametric points are involved (like endpoints of a chord), relationships between their parameters often simplify the algebra significantly.

• Circle (radius a, center origin): x = a cos θ, y = a sin θ
• Parabola y² = 4ax: x = at², y = 2at
• Ellipse x²/a² + y²/b² = 1: x = a cos θ, y = b sin θ
• Hyperbola x²/a² − y²/b² = 1: x = a sec θ, y = b tan θ (or x = a cosh t, y = b sinh t)
• Tangent to parabola at (at², 2at): ty = x + at²
• Tangent to ellipse at (a cos θ, b sin θ): (x cos θ)/a + (y sin θ)/b = 1
• Normal to parabola at (at², 2at): y + tx = 2at + at³
• Tangent to hyperbola at (a sec θ, b tan θ): (x sec θ)/a − (y tan θ)/b = 1

• Confusing the eccentric angle θ with the actual polar angle from the center. For an ellipse, θ is defined via the auxiliary circle, not by tan⁻¹(y/x).
• Forgetting the sign conventions in hyperbola parametric equations. The parameter θ has restricted ranges depending on which branch you are on.
• Using the wrong tangent formula. The tangent at parametric point (at², 2at) is ty = x + at², not y = tx + at². Mixing up the form leads to wrong answers.
• Arithmetic errors when substituting parametric coordinates into the conic equation to verify a point lies on the curve. Always double-check by plugging back in.

Question. The normal to the parabola y² = 8x at the point (2, 4) meets the parabola again at point Q. Find the coordinates of Q.

Solved twice independently; both solutions agree.