If the normal to the ellipse $3 x^{2}+4 y^{2}=12$ at a point $P$ on it is parallel to the line, $2 x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then $P Q$ is equal to :
$\frac{\sqrt{221}}{2}$
$\frac{\sqrt{157}}{2}$
$\frac{\sqrt{61}}{2}$
$\frac{5 \sqrt{5}}{2}$
Correct Option: , 4
$3 x^{2}+4 y^{2}=12$
$x=2 \cos \theta, y=\sqrt{3} \sin \theta$
Let $\mathrm{P}(2 \cos \theta, \sqrt{3 \sin \theta})$
Equation of normal is $\frac{\mathrm{a}^{2} \mathrm{x}}{\mathrm{x}_{1}}-\frac{\mathrm{b}^{2} \mathrm{y}}{\mathrm{y}_{1}}=\mathrm{a}^{2}-\mathrm{b}^{2}$
$2 x \sin \theta-\sqrt{3} \cos \theta y=\sin \theta \cos \theta$
Slope $\frac{2}{\sqrt{3}} \tan \theta=-2 \quad \therefore \tan \theta=-\sqrt{3}$
Equation of tangent is
it passes through $(4,4)$
$3 x \cos \theta+2 \sqrt{3} \sin \theta y=6$
$12 \cos \theta+8 \sqrt{3} \sin \theta=6$
$\cos \theta=-\frac{1}{2}, \sin \theta=\frac{\sqrt{3}}{2} \therefore \theta=120^{\circ}$
Hence point $\mathrm{P}$ is $\left(2 \cos 120^{\circ}, \sqrt{3} \sin 120^{\circ}\right)$
$\mathrm{P}\left(-1, \frac{3}{2}\right), \mathrm{Q}(4,4)$
$\mathrm{PQ}=\frac{5 \sqrt{5}}{2}$
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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,