Question
Find the length of the cycloid and show that the line
divides it in the ratio 1 : 3.
b) Find the condition for the curves, and
intersecting orthogonally.
Answer :
Word Count : 525
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The parametric equations of the cycloid are: [ x = \alpha (\theta - \sin \theta), \quad y = \alpha (1 - \cos \theta) ] The length of a curve given parametrically by (x(\theta), y(\theta)) is: [ L = \int \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}, d\theta ] Compute the derivatives: [ \frac{dx}{d\theta} = \alpha (1 - \cos \theta), \quad \frac{dy}{d\theta} = \alpha \sin \theta ] Then: [ \left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = \alpha^2 (1 - \cos \theta)^2 + \alpha^2 \sin^2 \theta = \alpha^2 \big((1 - \cos \theta)^2 + \sin^2 \theta\big) ] Simplify using (\sin^2 \theta + \cos^2 \theta = 1): [ (1 - \cos \theta)^2 + \sin^2 \theta = 1 - 2\cos \theta + \cos^2 \theta + \sin^2 \theta = 2 - 2\cos \theta = 4 \sin^2 \frac{\theta}{2} ] So: [ \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2} = \alpha \cdot 2 \sin \frac{\theta}{2} = 2\alpha \sin \frac{\theta}{2} ] Hence,
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Related Questions
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Find the length of the cycloid and show that the line
divides it in the ratio 1 : 3.
b) Find the condition for the curves, and
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