Find the The Slopes of the tangent and the normal to the following curves at the indicated points:
$x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$ at $\theta=\pi / 4$
Given:
$x=\operatorname{acos}^{3} \theta \& y=a \sin ^{3} \theta$ at $\theta=\frac{\pi}{4}$
Here, To find $\frac{d y}{d x}$, we have to find $\frac{d y}{d \theta} \& \frac{d x}{d \theta}$ and and divide $\frac{\frac{d y}{d \theta}}{d \theta}$ and we get our desired $\frac{d y}{d x}$.
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \cdot \mathrm{x}^{\mathrm{n}-1}$
$\Rightarrow \mathrm{x}=\mathrm{a} \cos ^{3} \theta$
$\Rightarrow \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}\left(\frac{\mathrm{dx}}{\mathrm{d} \theta}\left(\cos ^{3} \theta\right)\right)$
$\therefore \frac{\mathrm{d}}{\mathrm{dx}}(\cos \mathrm{x})=-\sin \mathrm{x}$
$\Rightarrow \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}\left(3 \cos ^{3}-1 \theta \times-\sin \theta\right)$
$\Rightarrow \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}\left(3 \cos ^{2} \theta \times-\sin \theta\right)$
$\Rightarrow \frac{\mathrm{dx}}{\mathrm{d} \theta}=-3 \mathrm{a} \cos ^{2} \theta \sin \theta \ldots(1)$
$\Rightarrow y=a \sin ^{3} \theta$
$\Rightarrow \frac{d y}{d \theta}=a\left(\frac{d y}{d \theta}\left(\sin ^{3} \theta\right)\right)$
$\therefore \frac{d}{d x}(\sin x)=\cos x$
$\Rightarrow \frac{d y}{d \theta}=a\left(3 \sin ^{3}-1 \theta \times \cos \theta\right)$
$\Rightarrow \frac{d y}{d \theta}=a\left(3 \sin ^{2} \theta \times \cos \theta\right)$
$\Rightarrow \frac{d y}{d \theta}=3 a \sin ^{2} \theta \cos \theta \ldots(2)$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}=\frac{-3 \operatorname{acos}^{2} \theta \sin \theta}{3 \operatorname{asin}^{2} \theta \cos \theta}$
$\Rightarrow \frac{d y}{d x}=\frac{-\cos \theta}{\sin \theta}$
$\Rightarrow \frac{d y}{d x}=-\tan \theta$
The Slope of the tangent is $-\tan \theta$
$\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\theta=\frac{\pi}{4}}=-\tan \left(\frac{\pi}{4}\right)$
$\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\theta=\frac{\pi}{4}}=-1$
$\therefore \tan \left(\frac{\pi}{4}\right)=1$
$\therefore$ The Slope of the tangent at $x=\frac{\pi}{4}$ is $-1$
$\Rightarrow$ The Slope of the normal $=\frac{-1}{\text { The Slope of the tangent }}$
$\Rightarrow$ The Slope of the normal $=\frac{-1}{\left(\frac{d y}{d x}\right)_{\theta=\frac{\pi}{4}}}$
$\Rightarrow$ The Slope of the normal $=\frac{-1}{-1}$
$\Rightarrow$ The Slope of the normal $=1$
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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,