Find the equation of the line drawn through the point of intersection of the lines x – y = 7 and 2x + y = 2 and passing through the origin.
Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right)$. Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations.
$x-y=7 \ldots$ (i)
$2 x+y=2 \ldots$ (ii)
Now, we find the point of intersection of eq. (i) and (ii)
Multiply the eq. (i) by 2, we get
$2 x-2 y=14 \ldots$ (iii)
On subtracting eq. (iii) from (ii), we get
$2 x-2 y-2 x-y=14-2$
$\Rightarrow-3 y=12$
$\Rightarrow y=-4$
Putting the value of y in eq. (i), we get
$x-(-4)=7$
$\Rightarrow x+4=7$
$\Rightarrow x=7-4$
$\Rightarrow x=3$
Hence, the point of intersection $\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ is $(3,-4)$
Let $A B$ is the line drawn from the point of intersection $(3,-4)$ and passing through the origin.
Firstly, we find the slope of the line joining the points $(3,-4)$ and $(0,0)$
Slope of line joining two points $=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
$\therefore \mathrm{m}_{\mathrm{AB}}=\frac{0-(-4)}{0-3}=\frac{4}{-3}$
Now, we have to find the equation of the line passing through the origin
Equation of line: $y-y_{1}=m\left(x-x_{1}\right)$
$\Rightarrow \mathrm{y}-0=-\frac{4}{3}(\mathrm{x}-0)$
$\Rightarrow 3 \mathrm{y}=-4 \mathrm{x}$
$\Rightarrow 4 \mathrm{x}+3 \mathrm{y}=0$
Hence, the equation of the line passing through the origin is $4 x+3 y=0$
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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,