Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • Find the equations of the tangent and normal to the given curves at the indicated points: x = cos t, y = sint at t = pi /4

14) Find the equations of the tangent and normal to the given curves at the indicated points:

e) x = \cos t , y = \sin t \: \: at \: \: t = \pi /4

Answers (1)

best_answer

We know that Slope of the tangent at a point on the given curve is given  by  \frac{dy}{dx}
Given the equation of the curve
x = \cos t , y = \sin t
Now,
\frac{dx}{dt} = -\sin t            and           \frac{dy}{dt} = \cos t
Now, 
\left ( \frac{dy}{dx} \right )_{t=\frac{\pi}{4}} = \frac{ \frac{dy}{dt}}{ \frac{dx}{dt}} = \frac{\cos t}{-\sin t} = -\cot t = =- \cot \frac{\pi}{4} = -1
Hence slope of the tangent is -1
Now we know that,
slope \ of \ normal = \frac{-1}{slope \ of \ tangent} = \frac{-1}{-1} = 1
Now, the equation of the tangent at the point t = \frac{\pi}{4} with slope = -1 is
x= \cos \frac{\pi}{4} = \frac{1}{\sqrt2}          and       

    y= \sin \frac{\pi}{4} = \frac{1}{\sqrt2}
equation of the tangent at

 t = \frac{\pi}{4} i.e. \left ( \frac{1}{\sqrt2}, \frac{1}{\sqrt2}\right ) is


y- y_1 = m(x-x_1)\\ y-\frac{1}{\sqrt2} = -1(x- \frac{1}{\sqrt2})\\ \sqrt2y + \sqrt2x = 2\\ y + x = \sqrt2
Similarly, the equation of normal at t = \frac{\pi}{4}  with slope = 1 is
x= \cos \frac{\pi}{4} = \frac{1}{\sqrt2}          and         

  y= \sin \frac{\pi}{4} = \frac{1}{\sqrt2}
equation of the tangent at

 t = \frac{\pi}{4} i.e. \left ( \frac{1}{\sqrt2}, \frac{1}{\sqrt2}\right ) is
\\y- y_1 = m(x-x_1)\\ y-\frac{1}{\sqrt2} = 1(x- \frac{1}{\sqrt2})\\ \sqrt2y - \sqrt2x = 0\\ y - x = 0\\ x=y
 

Posted by

Gautam harsolia

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Date Sheet 2025 (Out) Live: Class 10, 12 board exam time table at cbse.gov.in; syllabus, sample papers
anam-khan

CBSE Class 12 board exam date sheet 2025 out; exams from February 15 to April 4
anam-khan

When will CBSE Board exam 2025 date sheet be released for Class 10, 12?

Latest Question