The coordinates of the point P on the curve x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the parametric equations of the curve are $x=a(	heta+\sin 	heta)$ and $y=a(1-\cos 	heta)$.To find the slope of the tangent,we differentia

Vedclass · Accepted Answer

$\left[a\left(\frac{\pi}{2}+1\right), a\right]$

Vedclass · Answer

(B) Given the parametric equations of the curve are $x=a(	heta+\sin 	heta)$ and $y=a(1-\cos 	heta)$.To find the slope of the tangent,we differentiate $x$ and $y$ with respect to $	heta$:$\frac{dx}{d	heta} = a(1+\cos 	heta)$ and $\frac{dy}{d	heta} = a\sin 	heta$.Thus,the slope of the tangent is $\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a\sin 	heta}{a(1+\cos 	heta)} = \frac{2\sin(	heta/2)\cos(	heta/2)}{2\cos^2(	heta/2)} = 	an(	heta/2)$.Given that the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis,the slope is $	an(\frac{\pi}{4}) = 1$.Equating the slopes: $	an(	heta/2) = 	an(\frac{\pi}{4})$,which implies $	heta/2 = \frac{\pi}{4}$,so $	heta = \frac{\pi}{2}$.Substituting $	heta = \frac{\pi}{2}$ into the original equations:$x = a(\frac{\pi}{2} + \sin(\frac{\pi}{2})) = a(\frac{\pi}{2} + 1)$$y = a(1 - \cos(\frac{\pi}{2})) = a(1 - 0) = a$.Therefore,the coordinates of point $P$ are $\left[a\left(\frac{\pi}{2}+1ight), aight]$.

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$,where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis,are

Similar Questions

The polar equation of the line perpendicular to the line $\sin \theta - \cos \theta = \frac{1}{r}$ and passing through the point $\left(2, \frac{\pi}{6}\right)$ is

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \, years$ and $810 \, years$. The time, in years, after which one-fourth of the material remains is

Two waves $Y_1 = 0.25 \sin 316 t$ and $Y_2 = 0.25 \sin 310 t$ are propagating in the same direction. The number of beats produced per second is

The major product in the following reaction is:

Dicotyledons like mango,brinjal and monocotyledons like wheat and rice are placed under a taxonomic category known as

Vedclass Test Series

Exam Paper Generator

Online Exam Module