If the parabolas $y^{2}=4 b(x-c)$ and $y^{2}=8 a x$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$
$(1,1,0)$
$\left(\frac{1}{2}, 2,3\right)$
$\left(\frac{1}{2}, 2,0\right)$
$(1,1,3)$
Correct Option: 1, 2, 3, 4
Normal to these two curves are
$y=m(x-c)-2 b m-b m^{3}$
$y=m x-4 a m-2 a m^{3}$
If they have a common normal
$(c+2 b) m+b m^{3}=4 a m+2 a m^{3}$
Now $(4 a-c-2 b) m=(b-2 a) m^{3}$
We get all options are correct for $m=0$ (common normal x-axis)
Ans. $(1),(2),(3),(4)$
If we consider question as
If the parabolas $y^{2}=4 b(x-c)$ and $y^{2}=8 a x$ have a common normal other than $x$-axis, then which one of the following is a valid choice for the ordered triad $(a, b, c)$ ?
When $m \neq 0:(4 a-c-2 b)=(b-2 a) m^{2}$
$m^{2}=\frac{c}{2 a-b}-2>0 \Rightarrow \frac{c}{2 a-b}>2$
Now according to options, option 4 is correct
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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,