# If $x = a{t^2}$ and $y = 2at$, then $\dfrac{{{d^2}y}}{{d{x^2}}}$ at $t = \dfrac{1}{2}$ is

A.$\dfrac{{ - 2}}{a}$

B.$\dfrac{4}{a}$

C.$\dfrac{8}{a}$

D.$\dfrac{{ - 4}}{a}$

Last updated date: 25th Mar 2023

•

Total views: 311.7k

•

Views today: 5.88k

Answer

Verified

311.7k+ views

Hint : Use differentiation of parametric form method

The given equations are $x = a{t^2}$and $y = 2at$

On differentiating $x$ with respect to $t$ we get,

$\dfrac{{dx}}{{dt}} = \dfrac{{d(a{t^2})}}{{dt}} = 2at$ ……(i)

$\dfrac{{dt}}{{dx}} = \dfrac{1}{{2at}}$ ……(ii)

Similarly differentiating $y$ with respect to $t$ we get,

$\dfrac{{dy}}{{dt}} = \dfrac{{d(2at)}}{{dt}} = 2a$ ……(iii)

To get $\dfrac{{dy}}{{dx}}$ divide (iii) by (i)

So, $\dfrac{{dy}}{{dx}} = \dfrac{1}{t}$

Double differentiating the above equation we get

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{t^2}}}\dfrac{{dt}}{{dx}}$

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{t^2}}}\left( {\dfrac{1}{{2at}}} \right) = - \dfrac{1}{{2a{t^3}}}$ (From (ii)) ……(iv)

We have been asked to find the value of

$\dfrac{{{d^2}y}}{{d{x^2}}}$ at $t = \frac{1}{2}$

On putting $t = \dfrac{1}{2}$ in equation (iv) we get,

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{2a{{\left( {\dfrac{1}{2}} \right)}^3}}} = - \dfrac{1}{{2a\left( {\dfrac{1}{8}} \right)}} = - \dfrac{4}{a}$

Hence the correct option is D.

Note :-In these type of question of finding double differentiation at a particular point where the equations are in parametric form, we first have to differentiate it with respect to the variable assigned then divide them to get $\dfrac{{dy}}{{dx}}$ then double differentiate it as done above, at last put the value of the variable provided to get the answer.

The given equations are $x = a{t^2}$and $y = 2at$

On differentiating $x$ with respect to $t$ we get,

$\dfrac{{dx}}{{dt}} = \dfrac{{d(a{t^2})}}{{dt}} = 2at$ ……(i)

$\dfrac{{dt}}{{dx}} = \dfrac{1}{{2at}}$ ……(ii)

Similarly differentiating $y$ with respect to $t$ we get,

$\dfrac{{dy}}{{dt}} = \dfrac{{d(2at)}}{{dt}} = 2a$ ……(iii)

To get $\dfrac{{dy}}{{dx}}$ divide (iii) by (i)

So, $\dfrac{{dy}}{{dx}} = \dfrac{1}{t}$

Double differentiating the above equation we get

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{t^2}}}\dfrac{{dt}}{{dx}}$

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{t^2}}}\left( {\dfrac{1}{{2at}}} \right) = - \dfrac{1}{{2a{t^3}}}$ (From (ii)) ……(iv)

We have been asked to find the value of

$\dfrac{{{d^2}y}}{{d{x^2}}}$ at $t = \frac{1}{2}$

On putting $t = \dfrac{1}{2}$ in equation (iv) we get,

$\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{2a{{\left( {\dfrac{1}{2}} \right)}^3}}} = - \dfrac{1}{{2a\left( {\dfrac{1}{8}} \right)}} = - \dfrac{4}{a}$

Hence the correct option is D.

Note :-In these type of question of finding double differentiation at a particular point where the equations are in parametric form, we first have to differentiate it with respect to the variable assigned then divide them to get $\dfrac{{dy}}{{dx}}$ then double differentiate it as done above, at last put the value of the variable provided to get the answer.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE