If the line y = √3x + k touches the circle x2 + y2 = 16, then find the value of k.
Since, the equation of a circle having centre $(h, k)$, having radius as $r$ units, is
$(x-h)^{2}+(y-k)^{2}=r^{2}$
$(x-0)^{2}+(y-0)^{2}=4^{2}$
Perpendicular Distance between a point $(0,0) \&$ the line
$y=\sqrt{3} x+k$ or $\sqrt{3} x-y+k=0$
Perpendicular Distance (Between a point and line) $=$
Whereas the point is $\left(x_{1}, y_{1}\right)$ and the line is expressed as $a x+b y+c=0$
$D=\frac{\{|\sqrt{3}(0)+(0)(-1)+k|\}}{\sqrt{(\sqrt{3})^{2}+1^{2}}}=\frac{\{|k|\}}{\sqrt{3+1}}=\frac{\{k\}}{\sqrt{4}}=\frac{k}{2} \quad \frac{\left|a x_{1}+b y_{1}+c\right|}{\sqrt{a^{2}+b^{2}}}$
$\frac{k}{2}=4$ (Radius $=4$, Given)
$k=8$
Hence, the required value of $k$ is $8 .$
Hence, the required value of k is 8.
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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,