If $x=a\left(1-\cos ^{3} \theta\right), y=a \sin ^{3} \theta$, Prove that $\frac{d^{2} y}{d x^{2}}=\frac{32}{27 a} a t \theta=\frac{\pi}{6}$
Idea of parametric form of differentiation:
If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$.
Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$
We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}$
Given,
$x=a\left(1-\cos ^{3} \theta\right) \ldots \ldots$ equation 1
$y=a \sin ^{3} \theta, \ldots \ldots .$ equation 2
to prove : $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{32}{27 \mathrm{a}}$ at $\theta=\frac{\pi}{6}$.
We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.
Let's find $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$
As $\frac{\mathrm{d}^{2} y}{\mathrm{dx}^{2}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$
So, lets first find $d y / d x$ using parametric form and differentiate it again.
$\frac{\mathrm{dx}}{\mathrm{d} \theta}=\frac{\mathrm{d}}{\mathrm{d} \theta} \mathrm{a}\left(1-\cos ^{3} \theta\right)=3 \operatorname{acos}^{2} \theta \sin \theta \ldots \ldots$ equation 3 [using chain rule]
Similarly,
$\frac{\mathrm{dy}}{\mathrm{d} \theta}=\frac{\mathrm{d}}{\mathrm{d} \theta} \mathrm{a} \sin ^{3} \theta=3 \mathrm{a} \sin ^{2} \theta \cos \theta \ldots \ldots$ equation 4
$\left[\because \frac{d}{d x} \cos x=-\sin x \& \frac{d}{d x} \cos x=\sin x\right]$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}=\frac{3 \operatorname{asin}^{2} \theta \cos \theta}{3 \operatorname{acos}^{2} \theta \sin \theta}=\tan \theta$
Differentiating again w.r.t $\mathrm{x}$ :
$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\frac{\mathrm{d}}{\mathrm{dx}}(\tan \theta)$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\sec ^{2} \theta \frac{\mathrm{d} \theta}{\mathrm{dx}} \ldots . .$ equation 5
[ using chain rule and $\left.\frac{d}{d x} \tan x=\sec ^{2} x\right]$
From equation 3:
$\frac{d x}{d \theta}=3 \operatorname{acos}^{2} \theta \sin \theta$
$\therefore \frac{\mathrm{d} \theta}{\mathrm{dx}}=\frac{1}{3 \operatorname{acos}^{2} \theta \sin \theta}$
Putting the value in equation 5 :
$\frac{d^{2} y}{d x^{2}}=\sec ^{2} \theta \frac{1}{3 \operatorname{acos}^{2} \theta \sin \theta}$
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{1}{3 \mathrm{a} \cos ^{4} \theta \sin \theta}$
Put $\theta=\pi / 6$
$\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)$ at $\left(\mathrm{x}=\frac{\pi}{6}\right)=\frac{1}{3 \mathrm{a} \cos ^{4} \frac{\pi}{6} \sin \frac{\pi}{6}}=\frac{1}{3 \mathrm{a}\left(\frac{\sqrt{3}}{2}\right)^{4} \frac{1}{2}}$
$\therefore\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)$ at $\left(\mathrm{x}=\frac{\pi}{6}\right)=\frac{32}{27 \mathrm{a}} \ldots$ proved
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,