The point on the curve $y^{2}=4 x$ which is nearest to, the point $(2,1)$ is
(a) $1,2 \sqrt{2}$
(b) $(1,2)$
(c) $(1,-2)$
(d) $(-2,1)$
$(b)(1,2)$
Let the required point be $(x, y)$. Then,
$y^{2}=4 x$
$\Rightarrow x=\frac{y^{2}}{4}$ ......(1)
Now,
$d=\sqrt{(x-2)^{2}+(y-1)^{2}}$]
Squaring both sides, we get
$\Rightarrow d^{2}=(x-2)^{2}+(y-1)^{2}$
$\Rightarrow d^{2}=\left(\frac{y^{2}}{4}-2\right)^{2}+(y-1)^{2}$
$\Rightarrow d^{2}=\frac{y^{4}}{16}+4-y^{2}+y^{2}+1-2 y$ [From eq. (1)]
Now,
$Z=d^{2}=\frac{y^{4}}{16}+4-y^{2}+y^{2}+1-2 y$
$\Rightarrow \frac{d Z}{d y}=\frac{y^{3}}{4}-2 y+2 y-2$
$\Rightarrow \frac{d Z}{d y}=\frac{y^{3}}{4}-2$
$\Rightarrow \frac{y^{3}}{4}-2=0$
$\Rightarrow y^{3}=8$
$\Rightarrow y=2$
Substituting the value of $y$ in $(1)$, we get
$x=1$
Now,
$\frac{d^{2} Z}{d y^{2}}=\frac{3 y^{2}}{4}$
$\Rightarrow \frac{d^{2} Z}{d y^{2}}=\frac{3(2)^{2}}{4}=3>0$
So, the nearest point is $(1,2)$.
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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,