A parametric curve: x = a(θ + sin θ), y = a(1 − cos θ)प्राचलिक वक्र: x = a(θ + sin θ), y = a(1 − cos θ)Parametric curve: x = a(θ + sin θ), y = a(1 − cos θ)

NCERT Example 33 · Class 12 Maths · Chapter 5 · interactive diagram

Parts and what they mean

• x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
• y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
• Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.
• Curve traced by θ (here a = 1) (θ द्वारा बना वक्र (यहाँ a = 1)) — Each θ gives one point (x, y); θ is the parameter.
• Point at \theta = \frac{\pi}{2} (\theta = \frac{\pi}{2} पर बिंदु) — \frac{dy}{dx} = \tan\frac{\theta}{2} = \tan\frac{\pi}{4} = 1. Point \left(\frac{\pi}{2} + 1, 1\right).
• Tangent with slope 1 (ढाल 1 वाली स्पर्श रेखा) — Slope = \frac{dy/d\theta}{dx/d\theta} = \frac{\sin\theta}{1 + \cos\theta} at \theta = \frac{\pi}{2}.
• Sharp point at \theta = \pi (\theta = \pi पर नुकीला बिंदु) — Here \frac{dx}{d\theta} = 1 + \cos\pi = 0, so \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} cannot be used; the curve has a sharp point (cusp) here.
• Bottom point, \theta = 0 (निचला बिंदु, \theta = 0) — Slope \tan 0 = 0: the curve is flat here.

Original diagram drawn by Simple Learning Hub, based on NCERT Example 33. Report a mistake