Find the equation of the line through the intersection of the lines $5 x-3 y=1$ and $2 x+3 y=23$ and which is perpendicular to the line $5 x-3 y=1$
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.
$5 x-3 y=1 \ldots(i)$
$2 x+3 y=23 \ldots$ (ii)
Now, we find the point of intersection of eq. (i) and (ii)
Adding eq. (i) and (ii) we get
$5 x-3 y+2 x+3 y=1+23$
$\Rightarrow 7 x=24$
$\Rightarrow x=\frac{24}{7}$
Putting the value of x in eq. (i), we get
$5\left(\frac{24}{7}\right)-3 y=1$
$\Rightarrow \frac{120}{7}-3 \mathrm{y}=1$
$\Rightarrow-3 \mathrm{y}=1-\frac{120}{7}$
$\Rightarrow-3 \mathrm{y}=\frac{7-120}{7}$
$\Rightarrow-3 \mathrm{y}=-\frac{113}{7}$
$\Rightarrow \mathrm{y}=\frac{113}{21}$
Hence, the point of intersection $\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ is
$\left(\frac{24}{7}, \frac{113}{21}\right)$
Now, we know that, when two lines are perpendicular, then the product of their slope is equal to $-1$
$m_{1} \times m_{2}=-1$
$\Rightarrow$ Slope of the given line $\times$ Slope of the perpendicular line $=-1$
$\because \frac{5}{3} \times$ Slope of the perpendicular line $=-1$
$\Rightarrow$ The slope of the perpendicular line $=-\frac{3}{5}$
So, the slope of a line which is perpendicular to the given line is $-\frac{3}{5}$
Then the equation of the line passing through the point $\left(\frac{24}{7}, \frac{113}{21}\right)$ having slope
$-\frac{3}{5}$ is:
$y-y_{1}=m\left(x-x_{1}\right)$
$\Rightarrow \mathrm{y}-\left(\frac{113}{21}\right)=-\frac{3}{5}\left(\mathrm{x}-\frac{24}{7}\right)$
$\Rightarrow 5 \mathrm{y}-5 \times \frac{113}{21}=-3 \mathrm{x}+\frac{24}{7}$
$\Rightarrow 5 \mathrm{y}-\frac{565}{21}=-3 \mathrm{x}+\frac{72}{7}$
$\Rightarrow 3 \mathrm{x}+5 \mathrm{y}-\frac{565}{21}-\frac{72}{7}=0$
$\Rightarrow \frac{63 \mathrm{x}+105 \mathrm{y}-565-216}{21}=0$
$\Rightarrow 63 x+105 y-781=0$
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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,