If tangents are drawn to the ellipse $x^{2}+2 y^{2}=2$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted betwen the coordinate axes lie on the curve :
$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$
$\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$
$\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1$
$\frac{1}{4 x^{2}}+\frac{1}{2 y^{2}}=1$
Correct Option: , 3
Equation of general tangent on ellipse
$\frac{\mathrm{x}}{\mathrm{a} \sec \theta}+\frac{\mathrm{y}}{\mathrm{bcosec} \theta}=1$
$a=\sqrt{2}, b=1$
$\Rightarrow \frac{x}{\sqrt{2} \sec \theta}+\frac{y}{\operatorname{cosec} \theta}=1$
Let the midpoint be $(h, k)$
$\mathrm{h}=\frac{\sqrt{2} \sec \theta}{2} \Rightarrow \cos \theta=\frac{1}{\sqrt{2} \mathrm{~h}}$
and $k=\frac{\operatorname{cosec} \theta}{2} \Rightarrow \sin \theta=\frac{1}{2 k}$
$\because \sin ^{2} \theta+\cos ^{2} \theta=1$
$\Rightarrow \frac{1}{2 \mathrm{~h}^{2}}+\frac{1}{4 \mathrm{k}^{2}}=1$
$\Rightarrow \frac{1}{2 \mathrm{x}^{2}}+\frac{1}{4 \mathrm{y}^{2}}=1$
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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,