Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
Given lines are
2x + y = 5 ……1
x + 3y = -8 ……2
Firstly, we find the point of intersection of equation 1 and equation 2
Multiply the equation 2 by 2, we get
2x + 6y = -16 …….3
On subtracting equation 3 from 1, we get
2x + y – 2x – 6y = 5 – (-16)
On simplifying we get
⇒ -5y = 5 + 16
⇒ -5y = 21
$\Rightarrow y=-\frac{21}{5}$
Putting the value of $y$ in equation 1 , we get
$2 x+\left(-\frac{21}{5}\right)=5$
On rearranging we get
$\Rightarrow 2 x=5+\frac{21}{5}$
$\Rightarrow 2 \mathrm{x}=\frac{25+21}{5}$
$\Rightarrow 10 x=46$
$\Rightarrow \mathrm{x}=\frac{46}{10}=\frac{23}{5}$
Hence, the point of intersection is $\left(\frac{23}{5},-\frac{21}{5}\right)$
Now, we find the slope of the given equation $3 x+4 y=7$ We know that the slope of an equation is
$m=-a / b$
$\Rightarrow m=-\frac{3}{4}$
So, the slope of a line which is parallel to this line is also $-\frac{3}{4}$
Then the equation of the line passing through the point $\left(\frac{23}{5},-\frac{21}{5}\right)$ having
slope $^{-\frac{3}{4}}$ is:
$y-y_{1}=m\left(x-x_{1}\right)$
Substituting the values we get
$\Rightarrow y-\left(-\frac{21}{5}\right)=-\frac{3}{4}\left(x-\frac{23}{5}\right)$
Computing and simplifying
$\Rightarrow y+\frac{21}{5}=-\frac{3}{4} x+\frac{69}{20}$
$\Rightarrow \frac{3}{4} \mathrm{x}+\mathrm{y}=\frac{69}{20}-\frac{21}{5}$
$\Rightarrow \frac{3 x+4 y}{4}=\frac{69-84}{20}$
$\Rightarrow 3 x+4 y=-\frac{15}{5}$
$\Rightarrow 3 x+4 y+3=0$
$\Rightarrow \frac{3 x+4 y}{4}=\frac{69-84}{20}$
$\Rightarrow 3 x+4 y=-\frac{15}{5}$
$\Rightarrow 3 x+4 y+3=0$
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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,