Question:
If the line $y=x$ touches the curve $y=x^{2}+b x+c$ at a point $(1,1)$ then
A. $b=1, c=2$
B. $b=-1, c=1$
C. $b=2, c=1$
D. $b=-2, c=1$
Solution:
Given that line $y=x$ touches the curve $y=x^{2}+b x+c$ at a point $(1,1)$
Slope of line $=1$
Slope of tangent to the curve $=1$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}+\mathrm{b}$
$\Rightarrow 2 \mathrm{x}+\mathrm{b}=1$
$\Rightarrow 2+\mathrm{b}=1$
$\Rightarrow \mathrm{b}=-1$
Putting this and $x=1$ and $y=1$ in the equation of the curve,
$1=1-1+c$
$\Rightarrow c=1$
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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,
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