Find the equation of ellipse whose eccentricity is 2/3 , latus rectum is 5 and the centre is (0, 0).
We know that equation of an ellipse
$=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
Also we have,
Length of latus rectum $=\frac{2 b^{2}}{a}$
Length of minor $a x i s=2 b$
Given
$e=\frac{2}{3}$
$\mathrm{b}^{2}=\mathrm{a}^{2}\left(1-\mathrm{e}^{2}\right) \ldots \ldots \ldots 1$
Length of Latus Rectum $=\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=5$
$\mathrm{b}^{2}=2.5 \mathrm{a} \ldots \ldots \ldots \ldots .2$
Now by substituting the values in equation 1 we get
$\therefore 2.5 \mathrm{a}=\mathrm{a}^{2}\left(1-\left(\frac{2}{3}\right)^{2}\right)$
On simplifying and rearranging we get
$2.5 a=a^{2}\left(1-\frac{4}{9}\right)$
$2.5 a=a^{2}\left(\frac{5}{9}\right)$
$22.5 \mathrm{a}=5 \mathrm{a}^{2}$
$5 a^{2}-22.5 a=0$
$5 a(a-4.5)=0$
$a=0\{$ Not Possible, as the length of latus rectum is 5 units $\}$ or $4.5=9 / 2$ Now by substituting the values in equation 2 we get
$\mathrm{b}^{2}=2.5 \times 4.5=11.25=\frac{45}{4}$
Again we have equation of an ellipse
$=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
By substituting this values we get
$\frac{\mathrm{x}^{2}}{\left(\frac{9}{2}\right)^{2}}+\frac{\mathrm{y}^{2}}{\frac{45}{4}}=1$
$\frac{4 x^{2}}{81}+\frac{4 y^{2}}{45}=1$
$\therefore$ Equation of an ellipse is $\frac{4 x^{2}}{81}+\frac{4 y^{2}}{45}=1$
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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
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- Chemical Effects of Electric Current,