Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $9 y^{2}-4 x^{2}=36$
The given equation is $9 y^{2}-4 x^{2}=36$.
It can be written as
$9 y^{2}-4 x^{2}=36$
Or, $\frac{y^{2}}{4}-\frac{x^{2}}{9}=1$
$\mathrm{Or}, \frac{y^{2}}{2^{2}}-\frac{x^{2}}{3^{2}}=1$ .(1)
Or, $\frac{y^{2}}{2^{2}}-\frac{x^{2}}{3^{2}}=1$ (1)
On comparing equation (1) with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$, we obtain $a=2$ and $b=3$.
We know that $a^{2}+b^{2}=c^{2}$.
$\therefore c^{2}=4+9=13$
$\Rightarrow c=\sqrt{13}$
Therefore,
The coordinates of the foci are $(0, \pm \sqrt{13})$.
The coordinates of the vertices are $(0, \pm 2)$.
Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{13}}{2}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 9}{2}=9$
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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,