Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Given curves are equations of two circles,
xy = 4 ….. (i) and
x2 + y2 = 8 …. (ii)
Different equation (i) w.r.t., x
$x \cdot \frac{d y}{d x}+y \cdot 1=0$
$\Rightarrow \frac{d y}{d x}=-\frac{y}{x} \Rightarrow m_{1}=-\frac{y}{x}$ $\ldots($ iii $)$
where, $m_{1}$ is the slope of the tangent to the curve. Differentiating eq. (ii) w.r.t. $x$
$2 x+2 y \cdot \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=-\frac{x}{y} \Rightarrow m_{2}=-\frac{x}{y}$
where, $m_{2}$ is the slope of the tangent to the circle. To find the point of contact of the two circles
$m_{1}=m_{2} \Rightarrow-\frac{y}{x}=-\frac{x}{y} \Rightarrow x^{2}=y^{2}$
Putting the value of $y^{2}$ in eq. (ii)
$x^{2}+x^{2}=8 \Rightarrow 2 x^{2}=8 \Rightarrow x^{2}=4$
Thus, $\quad x=\pm 2$
And, $\quad x^{2}=y^{2} \Rightarrow y=\pm 2$
Therefore, the point of contact of the two circles are (2, 2) and (-2, 2).
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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,