Prove that the curves $x y=4$ and $x^{2}+y^{2}=8$ touch each other.
Given:
Curves $x y=4$ .....(1)
$\& x^{2}+y^{2}=8 \ldots(2)$
Solving $(1) \&(2)$, we get
$\Rightarrow x y=4$
$\Rightarrow x=\frac{4}{y}$
Substituting $x=\frac{4}{y}$ in $x^{2}+y^{2}=8$, we get,
$\Rightarrow\left(\frac{4}{y}\right)^{2}+y^{2}=8$
$\Rightarrow \frac{16}{y^{2}}+y^{2}=8$
$\Rightarrow 16+y^{4}=8 y^{2}$
$\Rightarrow y^{4}-8 y^{2}+16=0$
We will use factorization method to solve the above equation
$\Rightarrow y^{4}-4 y^{2}-4 y^{2}+16=0$
$\Rightarrow y^{2}\left(y^{2}-4\right)-4\left(y^{2}-4\right)=0$
$\Rightarrow\left(y^{2}-4\right)\left(y^{2}-4\right)=0$
$\Rightarrow y^{2}-4=0$
$\Rightarrow y^{2}=4$
$\Rightarrow y=\pm 2$
Substituting $y=\pm 2$ in $x=\frac{4}{y}$, we get,
$\Rightarrow x=\frac{4}{+2}$
$\Rightarrow x=\pm 2$
$\therefore$ The point of intersection of two curves $(2,2) \&$
$(-2,-2)$
First curve $x y=4$
$\Rightarrow 1 \times y+x \cdot \frac{d y}{d x}=0$
$\Rightarrow x \cdot \frac{d y}{d x}=-y$
$\Rightarrow m_{1}=\frac{-y}{x} \ldots(3)$
Second curve is $x^{2}+y^{2}=8$
Differentiating above w.r.t $x$,
$\Rightarrow 2 x+2 y \cdot \frac{d y}{d x}=0$
$\Rightarrow y \cdot \frac{d y}{d x}=-x$
$\Rightarrow m_{2}=\frac{d y}{d x}=\frac{-x}{y} \ldots(4)$
At $(2,2)$, we have,
$m_{1}=\frac{-y}{x}$
$\Rightarrow \frac{-2}{2}$
$m_{1}=-1$
At $(2,2)$, we have,
$\Rightarrow m_{2}=\frac{-x}{y}$
$\Rightarrow \frac{-2}{2}$
$\Rightarrow m_{2}=-1$
Clearly, $m_{1}=m_{2}=-1$ at $(2,2)$
So, given curve touch each other at $(2,2)$
- 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,
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,