Question:
Prove that:
(i) $\frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}=a b c$
(ii) $\left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b}$
Solution:
(i) Consider the left hand side:
$\frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}$
$=\frac{a+b+c}{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}}$
$=\frac{a+b+c}{\frac{c+a+b}{a b c}}$
$=(a+b+c) \times\left(\frac{a b c}{a+b+c}\right)$
$=a b c$
Therefore left hand side is equal to the right hand side. Hence proved.
(ii)
Consider the left hand side:
$\left(a^{-1}+b^{-1}\right)^{-1}$
$=\frac{1}{a^{-1}+b^{-1}}$
$=\frac{1}{\frac{1}{a}+\frac{1}{b}}$
$=\frac{1}{\frac{b+a}{a b}}$
$=\frac{a b}{a+b}$
Therefore left hand side is equal to the right hand side. Hence proved.
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- 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
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