Find the equation of the hyperbola having its foci at $(0, \pm \sqrt{14})$ and passing through the point P(3, 4).
Given: Foci at $(0, \pm \sqrt{14})$ and passing through the point $\mathrm{P}(3,4)$
Need to find: The equation of the hyperbola.
Let, the equation of the hyperbola be:
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
It passes through the point $P(3,4)$
So putting the values of $(x, y)$ we get,
$\frac{3^{2}}{a^{2}}-\frac{4^{2}}{b^{2}}=1 \Rightarrow \frac{9}{a^{2}}-\frac{16}{b^{2}}=1 \cdots(1)$
Foci at $(0, \pm \sqrt{14})$
So, ae $=\sqrt{14}$
We know, $e=\sqrt{1+\frac{b^{2}}{a^{2}}}$
$\Rightarrow a \sqrt{1+\frac{b^{2}}{a^{2}}}=\sqrt{14}$
$\Rightarrow \sqrt{a^{2}+b^{2}}=\sqrt{14}$
$\Rightarrow \mathrm{a}^{2}+\mathrm{b}^{2}=14$ [Squaring on both sides]
$\Rightarrow a^{2}=14-b^{2}$ .............(2)
Comparing (1) and (2) we get
$\frac{9}{14-b^{2}}-\frac{16}{b^{2}}=1$
$\frac{9}{14-b^{2}}=1+\frac{16}{b^{2}}=\frac{b^{2}+16}{b^{2}}$
$9 b^{2}=14 b^{2}-b^{4}+224-16 b^{2}$
$b^{4}+11 b^{2}-224=0$
Solving the equations we get,
$b_{1}=\sqrt{\frac{1}{2}(-11+3 \sqrt{113})}$
$b_{2}=-\sqrt{\frac{1}{2}(-11+3 \sqrt{113})}$
$b_{3}=(-i) \sqrt{\frac{1}{2}(11+3 \sqrt{113})}$
$b_{4}=i \sqrt{\frac{1}{2}(11+3 \sqrt{113})}$
With the help of any of these values of b we can't find out the equation of the hyperbola.
* This is the only process we can apply in this standard.
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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,