Find the point on the curve $y 2=4 x$ which is nearest to the point $(2,-8)$
Let point $(x, y)$ be the nearest to the point $(2,-8)$. Then,
$y^{2}=4 x$
$\Rightarrow x=\frac{y^{2}}{4}$ ......(1)
$d^{2}=(x-2)^{2}+(y+8)^{2}$ [Using distance formula]
Now,
$Z=d^{2}=(x-2)^{2}+(y+8)^{2}$
$\Rightarrow Z=\left(\frac{y^{2}}{4}-2\right)^{2}+(y+8)^{2}$ [From eq. (1)]
$\Rightarrow Z=\frac{y^{4}}{16}+4-y^{2}+y^{2}+64+16 y$
$\Rightarrow \frac{d Z}{d y}=\frac{4 y^{3}}{16}+16$
For maximum or minimum values of $Z$, we must have
$\frac{d Z}{d y}=0$
$\Rightarrow \frac{4 y^{3}}{16}+16=0$
$\Rightarrow \frac{4 y^{3}}{16}=-16$
$\Rightarrow y^{3}=-64$
$\Rightarrow y=-4$
Substituting the value of $x$ in eq. (1), we get
$x=4$
Now,
$\frac{d^{2} Z}{d y^{2}}=\frac{12 y^{2}}{16}$
$\Rightarrow \frac{d^{2} Z}{d y^{2}}=12>0$
So, the nearest point is $(4,-4)$.
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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,