Question 3
The tangent to the curve given by x = et . cost, y = et . sint at t =
𝜋/4 makes with x-axis an angle:
0 (B) 𝜋/4 (C) 𝜋/3 (D) 𝜋/2
Given curve
𝒙=𝒆^𝒕.𝒄𝒐𝒔𝒕 and 𝒚=𝒆^𝒕.𝒔𝒊𝒏𝒕
Finding slope of tangent
𝒅𝒚/𝒅𝒙=(𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕)
𝒙=𝒆^𝒕.𝒄𝒐𝒔𝒕
Differentiating 𝑥 𝑤.𝑟.𝑡 𝑡
𝑑𝑥/𝑑𝑡=𝑒^𝑡 𝑐𝑜𝑠𝑡−𝑒^𝑡 𝑠𝑖𝑛𝑡
𝒅𝒙/𝒅𝒕=𝑒^𝑡 (𝑐𝑜𝑠𝑡−𝑠𝑖𝑛𝑡)
𝒚=𝒆^𝒕.𝒔𝒊𝒏𝒕
Differentiating y 𝑤.𝑟.𝑡
𝑑𝑦/𝑑𝑡=𝑒^𝑡 𝑠𝑖𝑛𝑡+𝑒^𝑡 𝑐𝑜𝑠𝑡
𝒅𝒚/𝒅𝒕=𝑒^𝑡 (𝑠𝑖𝑛𝑡\+𝑐𝑜𝑠𝑡)
Now,
𝒅𝒚/𝒅𝒙=(𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕)
𝑑𝑦/𝑑𝑥=(𝑒^𝑡 (𝑠𝑖𝑛𝑡\+𝑐𝑜𝑠𝑡)" " )/(𝑒^𝑡 (𝑐𝑜𝑠𝑡−𝑠𝑖𝑛𝑡)" " )
𝑑𝑦/𝑑𝑥=( (𝒔𝒊𝒏𝒕\+𝒄𝒐𝒔𝒕)" " )/( (𝒄𝒐𝒔𝒕−𝒔𝒊𝒏𝒕)" " )
Putting 𝐭=𝛑/𝟒
├ 𝒅𝒚/𝒅𝒙┤|_(𝒕=𝝅/𝟒)=( (𝑠𝑖𝑛 𝜋/4 \+ 𝑐𝑜𝑠 𝜋/4)" " )/( (𝑐𝑜𝑠 𝜋/4 − 𝑠𝑖𝑛 𝜋/4)" " )
= (1/√2+1/√2)/(1/√2−1/√2)
= (2/(√2))/0
= ∞
Let 𝜽 be the angle made by tangent with the 𝑥- axis.
Slope = 𝐭𝐚𝐧𝜽
∞ = tan𝜃
tan〖𝜃=∞〗
∴ 𝜽= 𝝅/𝟐
So, the correct answer is (D)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!