Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.
Let the equation of the required ellipse is
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ............(i)
It is given that
Length of Latus Rectum $=\frac{1}{2}$ minor Axis
We know that,
Length of Latus Rectum $=\frac{2 b^{2}}{a}$
and Length of Minor Axis = 2b
So, according to the given condition,
$\frac{2 b^{2}}{a}=\frac{1}{2} \times 2 b$
$\Rightarrow \frac{2 b^{2}}{a}=b$
$\Rightarrow \frac{2 \mathrm{~b}^{2}}{\mathrm{~b}}=\mathrm{a}$
$\Rightarrow 2 \mathrm{~b}=\mathrm{a}$ …(ii)
Now, we have to find the eccentricity
We know that
Eccentricity, $\mathrm{e}=\frac{\mathrm{c}}{\mathrm{a}}$ …(iii)
where, $c^{2}=a^{2}-b^{2}$
So, $c^{2}=(2 b)^{2}-b^{2}[$ from (ii) $]$
$\Rightarrow c^{2}=4 b^{2}-b^{2}$
$\Rightarrow c^{2}=3 b^{2}$
$\Rightarrow c=\sqrt{3} b^{2}$
$\Rightarrow c=b \sqrt{3}$
Substituting the value of c and a in eq. (iii), we get
Eccentricity, $e=\frac{c}{a}$
$=\frac{b \sqrt{3}}{2 b}$
$\therefore \mathrm{e}=\frac{\sqrt{3}}{2}$
- 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,