At what point on the circle $x^{2}+y^{2}-2 x-4 y+1=0$, the tangent is parallel to $x$ - axis.
Given:
The curve is $x^{2}+y^{2}-2 x-4 y+1=0$
Differentiating the above w.r.t $x$
$\Rightarrow x^{2}+y^{2}-2 x-4 y+1=0$
$\Rightarrow 2 x^{2}-1+2 y^{2}-1 \times \frac{d y}{d x}-2-4 x \frac{d y}{d x}+0=0$
$\Rightarrow 2 x+2 y \frac{d y}{d x}-2-4 \frac{d y}{d x}=0$
$\Rightarrow \frac{\mathrm{d} y}{\mathrm{dx}}(2 y-4)=-2 x+2$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-2(\mathrm{x}-1)}{2(\mathrm{y}-2)}$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-(\mathrm{x}-1)}{(\mathrm{y}-2)} \ldots(1)$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=$ The Slope of the tangent $=\tan \theta$
Since, the tangent is parallel to $x$ - axis
i.e,
$\Rightarrow \frac{d y}{d x}=\tan (0)=0 \ldots(2)$
$\therefore \tan (0)=0$
From (1) & (2), we get,
$\Rightarrow \frac{-(x-1)}{(y-2)}=0$
$\Rightarrow-(x-1)=0$
$\Rightarrow x=1$
Substituting $x=1$ in $x^{2}+y^{2}-2 x-4 y+1=0$, we get,
$\Rightarrow 1^{2}+y^{2}-2 x 1-4 y+1=0$
$\Rightarrow 1-y^{2}-2-4 y+1=0$
$\Rightarrow y^{2}-4 y=0$
$\Rightarrow y(y-4)=0$
$\Rightarrow y=0 \& y=4$
Thus, the required point is $(1,0) \&(1,4)$
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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,