Show that the following set of curves intersect orthogonally:
$x^{3}-3 x y^{2}=-2$ and $3 x^{2} y-y^{3}=2$
Given:
Curves $x^{3}-3 x y^{2}=-2 \ldots$ (1)
$\& 3 x^{2} y-y^{3}=2 \ldots(2)$
Adding (1) \& (2), we get
$\Rightarrow x^{3}-3 x y^{2}+3 x^{2} y-y^{3}=-2+2$
$\Rightarrow x^{3}-3 x y^{2}+3 x^{2} y-y^{3}=-0$
$\Rightarrow(x-y)^{3}=0$
$\Rightarrow(x-y)=0$
$\Rightarrow x=y$
Substituting $x=y$ on $x^{3}-3 x y^{2}=-2$
$\Rightarrow x^{3}-3 x x \times x^{2}=-2$
$\Rightarrow x^{3}-3 x^{3}=-2$
$\Rightarrow-2 x^{3}=-2$
$\Rightarrow x^{3}=1$
$\Rightarrow x=1$
Since $x=y$
$y=1$
The point of intersection of two curves is $(1,1)$
First curve $x^{3}-3 x y^{2}=-2$
Differentiating above w.r.t $x$,
$\Rightarrow 3 x^{2}-3\left(1 \times y^{2}+x \times 2 y \frac{d y}{d x}\right)=0$
$\Rightarrow 3 x^{2}-3 y^{2}-6 x y \frac{d y}{d x}=0$
$\Rightarrow 3 x^{2}-3 y^{2}=6 x y \frac{d y}{d x}$
$\Rightarrow \frac{d y}{d x}=\frac{3 x^{2}-3 y^{2}}{6 x y}$
$\Rightarrow \frac{d y}{d x}=\frac{3\left(x^{2}-y^{2}\right)}{6 x y}$
$\Rightarrow m_{1}=\frac{\left(x^{2}-y^{2}\right)}{2 x y}$ .....(3)
Second curve $3 x^{2} y-y^{3}=2$
Differentiating above w.r.t $x$,
$\Rightarrow 3\left(2 x \times y+x^{2} \times \frac{d y}{d x}\right)-3 y^{2} \frac{d y}{d x}=0$
$\Rightarrow 6 x y+3 x^{2} \frac{d y}{d x}-3 y^{2} \frac{d y}{d x}=0$
$\Rightarrow 6 x y+\left(3 x^{2}-3 y^{2}\right) \frac{d y}{d x}=0$
$\Rightarrow \frac{d y}{d x}=\frac{-6 x y}{3 x^{2}-3 y^{2}}$
$\Rightarrow \frac{d y}{d x}=\frac{-2 x y}{x^{2}-y^{2}}$
$\Rightarrow m_{2}=\frac{-2 x y}{x^{2}-y^{2}} \ldots(4)$
When $m_{1}=\frac{\left(x^{2}-y^{2}\right)}{2 x y} \& m_{2}=\frac{-2 x y}{x^{2}-y^{2}}$
$\Rightarrow \frac{\left(x^{2}-y^{2}\right)}{2 x y} \times \frac{-2 x y}{x^{2}-y^{2}}=-1$
$\therefore$ Two curves $x^{3}-3 x y^{2}=-2 \& 3 x^{2} y-y^{3}=2$ intersect orthogonally.
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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,