Question

Show that the tangent at any point 'A' on the
curve x=csecA, y=cTanA is ysinA= x-c cosA​

Answer 1

Answer:

Step-by-step explanation:

Given:

x=at2

y=at4

Then,

⇒t3=ax​,t4=ay​

⇒t=(ax​)1/3t=(ay​)1/4

⇒(ax​)1/3=(ay​)1/4

⇒(ax​)31​×124​=(ay​)41​×12

⇒a4x4​=a2y3​

⇒y3=ax4​ at P(h,k)

Also relation k3=ah4​

Now by differentiation of after equation we get

3y2dxdy​=a4x2​

dxdy​=3ay24x3​

Now MT​=dxdy​=3ay24x3​

Now we will find slope of tangent

MT​/P(h,x)=3ax24h3​

equation of tangent P(h,k)

y−k=3ak24h3​(x−h)

⇒ Let y=0

Then, −k=3ak24h3​(x−h)

⇒4h3−3ak3​=x−h

Earlier we found that k3=ah4​

Then ax3=h4

4h2−3h4​=x−h

⇒4−3h​+h=x

⇒x=4h​

⇒hx​=41​