Home
Class 12
MATHS
If a curve is represented parametrically...

If a curve is represented parametrically by the equation `x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))`

Text Solution

Verified by Experts

`(dx)/(dy)=(1)/((dy)/(dx))`
`rArr" "(d^(2)x)/(dy^(2))=(d)/(dy)((1)/((dy)/(dx)))=(d)/(dx)((1)/((dy)/(dx))).(dx)/(dy)=(-1)/(((dy)/(dx))^(2)).((d^(2)y)/(dx^(2)))((dx)/(dy))`
`therefore" "(d^(2)x)/(dy^(2))=((-d^(2)y)/(dx^(2)))/(((dy)/(d))^(3))or (d^(2)y)/(dx^(2))=-((dy)/(dx))^(3).(d^(2)x)/(dy^(2))`
`rArr" "(d^(2)y)/(dx^(2)=-((dy//dt)/(dx//dt))^(3).(d^(2)x)/(dy^(2))`
`rArr" "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(-3)((d^(2)x)/(dy^(2)))`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE|Exercise Exercise 3.1|7 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Exercise 3.2|38 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Question Bank|25 Videos

  • DOT PRODUCT

    CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

If x=f(t) and y=g(t), then write the value of (d^(2)y)/(dx^(2))

If x=f(t) and y=g(t), then (d^(2)y)/(dx^(2)) is equal to

The focus of the conic represented parametrically by the equation y=t^(2)+3, x= 2t-1 is

If x=tlogt,y=t^(t)," then "(d^2y)/(dx^(2))=

If x=t^(3)+t+5 and y=sin t then (d^(2)y)/(dx^(2))

If a curve is represented parametrically by the equations x=4t^(3)+3 and y=4+3t^(4) and (d^(2)x)/(dy^(2))/((dx)/(dy))^(n) is constant then the value of n, is

A curve is represented paramtrically by the equations x=e^(t)cost and y=e^(t)sint where t is a parameter. Then The value of (d^(2)y)/(dx^(2)) at the point where t=0 is

If x=a sin t-b cos t,quad y=a cos t+b sin t prove that (d^(2)y)/(dx^(2))=-(x^(2)+y^(2))/(y^(3))