Parametric representation expresses a curve using a parameter t: x = f(t), y = g(t). This is natural for many curves like circles, ellipses, and cycloids.

Key Parametric Curves in JEE:

Circle: x = acos t, y = asin t. dy/dx = -cot t. d^2y/dx^2 = -cosec^3(t)/a.

Ellipse: x = acos t, y = bsin t. dy/dx = -(b/a)cot t. The tangent equation at parameter t: (xcos t)/a + (y*sin t)/b = 1.

Parabola: x = at^2, y = 2at. dy/dx = 1/t. The tangent equation at (at^2, 2at): ty = x + at^2.

Cycloid: x = a(t-sin t), y = a(1-cos t). dy/dx = sin t/(1-cos t) = cot(t/2). This curve is generated by a point on a rolling circle.

Astroid: x = acos^3(t), y = asin^3(t). dy/dx = -tan t. This is a hypocycloid with four cusps.

Involute of circle: x = a(cos t + tsin t), y = a(sin t - tcos t). dy/dx = tan t.

Second Derivative Computation:

1. Find dy/dx as a function of t
2. Differentiate dy/dx with respect to t to get d(dy/dx)/dt
3. Divide by dx/dt: d^2y/dx^2 = [d(dy/dx)/dt] / [dx/dt]

Common Error: Students write d^2y/dx^2 = (d^2y/dt^2)/(d^2x/dt^2). This is WRONG and will give incorrect answers. The correct formula treats dy/dx as a single entity to be differentiated with respect to t.