Question:
Let $S=\left\{(x, y) \in R^{2}: \frac{y^{2}}{1+r}-\frac{x^{2}}{1-r}=1\right\}, \quad$ where
$r \neq \pm 1$. Then $\mathrm{S}$ represents :
A hyperbola whose eccentricity is $\frac{2}{\sqrt{\mathrm{r}+1}}$,
where $0
An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$,
where $r>1$
An ellipse whose eccentricity is $\frac{1}{\sqrt{\mathrm{r}+1}}$,
where $r>1$
An ellipse whose eccentricity is $\sqrt{\frac{2}{\mathrm{r}+1}}$,
when $r>1$
Correct Option: , 4
Solution:
$\frac{\mathrm{y}^{2}}{1+\mathrm{r}}-\frac{\mathrm{x}^{2}}{1-\mathrm{r}}=1$
for $r>1, \quad \frac{y^{2}}{1+r}+\frac{x^{2}}{r-1}=1$
$e=\sqrt{1-\left(\frac{r-1}{r+1}\right)}$
$=\sqrt{\frac{(r+1)-(r-1)}{(r+1)}}$
$=\sqrt{\frac{2}{r+1}}=\sqrt{\frac{2}{r+1}}$
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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
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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