Conic sections: parametric forms save the algebra
Coordinate geometry is rarely hard in concept and frequently brutal in algebra. One parameter instead of two coordinates is the fix.
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Coordinate geometry questions in JEE are rarely hard in concept and frequently brutal in algebra. Parametric forms are how you avoid the algebra: a general point carrying one parameter, instead of two coordinates plus a constraint, turns most conic problems from simultaneous equations into a single-variable exercise.
The four you need
Write them out until they are automatic. The time saved is not in recalling them, it is in the algebra they prevent.Where they pay off
Chords through two points. The chord joining parameters and on a parabola has a clean form, and the condition for it to be a focal chord is simply . Compare that with solving a quadratic in two unknowns. Tangents and normals. The tangent at the parametric point is a one-line result for each conic. The intersection of two tangents, or the locus of that intersection, becomes elimination between two parameters instead of a system in and . Locus problems. This is the big one. Write the moving point in terms of the parameter, then eliminate the parameter. Almost every locus question in the chapter is that two-step recipe, and almost every failed attempt is someone working directly in Cartesian coordinates. Focal distances. For a parabola, the focal distance of the point is . Sums and harmonic means of focal distances, which look like they need work, then come out immediately.The trap
Parametric forms hide domain restrictions. The secant-and-tangent parametrisation of a hyperbola covers only one branch for a given range of , and the secant is undefined at . When a question asks for the number of solutions, or for points on both branches, go back to Cartesian or handle the branches separately. Losing a solution here is a common and expensive mistake.A drill
Take fifteen conic questions and, before solving, write only the parametrisation you would use and the quantity you would eliminate. That decision determines whether the question takes four minutes or fourteen. Everything after it is routine.Tags
- coordinate geometry
- conic sections
- exam technique