If the points $A(a, 0), B(0, b)$ and $P(x, y)$ are collinear, using slopes, prove that
$\frac{x}{a}+\frac{y}{b}=1$
Given points are A(a,0),B(0,b) and P(x,y)
For three points to be collinear, the slope of all pairs must be equal, that is the slope of AB = slope of BP = slope of PA.
slope $=\left(\frac{\mathrm{y}_{2}-\mathrm{y}_{1}}{\mathrm{x}_{2}-\mathrm{x}_{1}}\right)$
Slope of $A B=\left(\frac{b-0}{0-a}\right)=\frac{b}{-a}$
Slope of BP $=\left(\frac{y-b}{x-0}\right)=\frac{y-b}{x}$
Slope of PA $=\left(\frac{y-0}{x-a}\right)=\frac{y}{x-a}$
Now Slope of AB = BP = PA
$\frac{b}{-a}=\frac{y-b}{x}=\frac{y}{x-a}$
Using the first two equality
$\Rightarrow \frac{\mathrm{b}}{-\mathrm{a}}=\frac{\mathrm{y}-\mathrm{b}}{\mathrm{x}}$
$\Rightarrow \mathrm{bx}=-\mathrm{a}(\mathrm{y}-\mathrm{b})$
$\Rightarrow \mathrm{bx}=-\mathrm{ay}+\mathrm{ab}$
Dividing the equation by “ab”, We get
$\frac{\mathrm{x}}{\mathrm{a}}=-\frac{\mathrm{y}}{\mathrm{b}}+1$
$\Rightarrow \frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$
Hence proved.
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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,