Find the equation of the lines through the point (3, 2) which make an angle of $45^{\circ}$ with the line $x-2 y=3$.
Let the slope of the required line be m1.
The given line can be written as $y=\frac{1}{2} x-\frac{3}{2}$, which is of the form $y=m x+c$
$\therefore$ Slope of the given line $=m_{2}=\frac{1}{2}$
It is given that the angle between the required line and line $x-2 y=3$ is $45^{\circ}$.
We know that if $\theta$ isthe acute angle between lines $I_{1}$ and $I_{2}$ with slopes $m_{1}$ and $m_{2}$ respectively, then $\tan \theta=\left|\frac{m_{2}-m_{1}}{1+m_{1} m_{2}}\right|$.
$\therefore \tan 45^{\circ}=\frac{\left|m_{1}-m_{2}\right|}{1+m_{1} m_{2}}$
$\Rightarrow 1=\left|\begin{array}{l}\frac{1}{2}-m_{1} \\ 1+\frac{m_{1}}{2}\end{array}\right|$
$\Rightarrow 1=\left|\frac{\left(\frac{1-2 m_{1}}{2}\right)}{\frac{2+m_{1}}{2}}\right|$
$\Rightarrow 1=\left|\frac{1-2 m_{1}}{2+m_{1}}\right|$
$\Rightarrow 1=\pm\left(\frac{1-2 m_{1}}{2+m_{1}}\right)$
$\Rightarrow 1=\frac{1-2 m_{1}}{2+m_{1}}$ or $1=-\left(\frac{1-2 m_{1}}{2+m_{1}}\right)$
$\Rightarrow 2+m_{1}=1-2 m_{1}$ or $2+m_{1}=-1+2 m_{1}$
$\Rightarrow m_{1}=-\frac{1}{3}$ or $m_{1}=3$
Case I: m1 = 3
The equation of the line passing through (3, 2) and having a slope of 3 is:
y – 2 = 3 (x – 3)
y – 2 = 3x – 9
3x – y = 7
Case II: $m_{1}=-\frac{1}{3}$
The equation of the line passing through $(3,2)$ and having a slope of $-\frac{1}{3}$ is:
$y-2=-\frac{1}{3}(x-3)$
$3 y-6=-x+3$
$x+3 y=9$
Thus, the equations of the lines are 3x – y = 7 and x + 3y = 9.
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- 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,