A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1 : 2. Find the equation of the line.
Given: A line perpendicular to the line segment joining the points $(1,0)$ and $(2,3)$ divides it in the ratio $1: 2$.
Formula to be used: If $(a, b)$ and $(c, d)$ are two points then the equation of the line passing through them is
$\frac{\mathrm{y}-\mathrm{b}}{\mathrm{x}-\mathrm{a}}=\frac{\mathrm{b}-\mathrm{d}}{\mathrm{a}-\mathrm{c}}$
If $\left(a_{1}, b_{1}\right)$ and $\left(a_{2}, b_{2}\right)$ be two points, then the co - ordinates of the point dividing their join in the ratio $a: b$ is given by
$\mathrm{x}-\mathrm{co}$ ordinate $=\left(\frac{\mathrm{a}_{1} \mathrm{Xb}+\mathrm{a}_{2} \mathrm{Xa}}{\mathrm{a}+\mathrm{b}}\right)$
$\mathrm{y}-\mathrm{co}$ ordinate $=\left(\frac{\mathrm{b}_{1} \mathrm{Xb}+\mathrm{b}_{2} \mathrm{Xa}}{\mathrm{a}+\mathrm{b}}\right)$
The equation of the line joining points (1,0) and (2,3) is
$\frac{y-0}{x-1}=\frac{0-3}{1-2}$
or, $\frac{y}{x-1}=\frac{-3}{-1}=3$
or,
$y=3 x-3$ or, $3 x-y=3$
i.e. the given line is 3x - y = 3.
Accordingly, the required co - ordinates of the point dividing the join of $(1,0)$ and $(2,3)$ in the ratio $1: 2$ are
$\left(\left(\frac{1 X 2+2 X 1}{1+2}\right),\left(\frac{0 X 2+3 X 1}{1+2}\right)\right)=\left(\frac{4}{3}, 1\right)$
The given line is $3 x-y=3$.
The slope of this line is 3 .
$\therefore$ the slope of the perpendicular line $=\frac{-1}{3}=-\frac{1}{3}$.
The equation of the line can be written in the form $\mathrm{y}=-\frac{1}{3} \mathrm{x}+\mathrm{c}$
( $c$ is the $y$-intercept)
This line will pass through $\left(\frac{4}{3}, 1\right)$.
$\therefore 1=-\frac{1}{3} \mathrm{X} \frac{4}{3}+\mathrm{c}$ or, $\mathrm{c}=1+\frac{4}{9}=\frac{13}{9}$
The required equation is $y=-\frac{1}{3} x+\frac{13}{9}$
i.e. $3 x+9 y=13$
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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,