Question

If the parametric equation of the curve is given by x=ae^t sin⁡t & y=ae^t cos⁡ t the value of \(\frac{ds}{dt}\) is given by _____

(a) ae^t

(b) 2ae^t

(c) √3 ae^t

(d) √2 ae^t

I got this question by my school principal while I was bunking the class.

My doubt stems from Derivative of Arc Length topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right option is (d) √2 ae^t

For explanation I would say: w.k.t \(\frac{ds}{dt} = \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}…(1)\)

\(\frac{dx}{dt} = ae^t (cos t + sin t), \frac{dy}{dt} = ae^t (-sin⁡ t + cos⁡t)\)

substituting in (1) we get

\(\frac{ds}{dt} = ae^t \sqrt{(cos t + sin t)^2+(-sin t+cos⁡t)^2}\)

\(\frac{ds}{dt} = ae^t \sqrt{1+2 sin t cos + 1 – 2 sin⁡ t cos⁡t} = \sqrt{2} ae^t…(cos^{2} t + sin{2} t = 1).\)