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$$\frac{t}{a}.

Ellipse: x = acos t, y = bsin t. dy/dx = -$\frac{b}{a}$*cot t. The tangent equation at parameter t: $\frac{x*cos t}{a}$ + $\frac{y*sin t}{b}$ = 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 $\frac{t}{1-cos t}$ = cot$\frac{t}{2}$. This curve is generated by a point on a rolling circle.

Astroid: x = a*$cos^3$(t), y = a*$sin^3$(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$\frac{dy/dx}{dt}$
3. Divide by dx/dt: $d^{2y}$/$dx^2$ = [d$\frac{dy/dx}{dt}$] / [dx/dt]

Common Error: Students write $d^{2y}$/$dx^2$ = $\frac{d^2y/dt^2}{(d^2x/dt^2)}$. 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.