Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $4 x^{2}+9 y^{2}=36$
The given equation is $4 x^{2}+9 y^{2}=36$.
It can be written as
$4 x^{2}+9 y^{2}=36$
Or, $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
$\mathrm{Or}, \frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}=1$ $\ldots(1)$
Here, the denominator of $\frac{x^{2}}{3^{2}}$ is greater than the denominator of $\frac{y^{2}}{2^{2}}$.
Therefore, the major axis is along the $x$-axis, while the minor axis is along the $y$-axis.
On comparing the given equation with $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, we obtain $a=3$ and $b=2$.
$\therefore c=\sqrt{a^{2}-b^{2}}=\sqrt{9-4}=\sqrt{5}$
Therefore,
The coordinates of the foci are $(\pm \sqrt{5}, 0)$.
The coordinates of the vertices are (±3, 0).
Length of major axis = 2a = 6
Length of minor axis = 2b = 4
Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{5}}{3}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 4}{3}=\frac{8}{3}$
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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,