Two lines passing through the point $(2,3)$ intersects each other at an angle of $60^{\circ}$. If slope of one line is 2 , find equation of the other line.
It is given that the slope of the first line, $m_{1}=2$.
Let the slope of the other line be $m_{2}$.
The angle between the two lines is $60^{\circ}$.
$\therefore \tan 60^{\circ}=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$
$\Rightarrow \sqrt{3}=\left|\frac{2-m_{2}}{1+2 m_{2}}\right|$
$\Rightarrow \sqrt{3}=\pm\left(\frac{2-m_{2}}{1+2 m_{2}}\right)$
$\Rightarrow \sqrt{3}=\frac{2-m_{2}}{1+2 m_{2}}$ or $\sqrt{3}=-\left(\frac{2-m_{2}}{1+2 m_{2}}\right)$
$\Rightarrow \sqrt{3}\left(1+2 m_{2}\right)=2-m_{2}$ or $\sqrt{3}\left(1+2 m_{2}\right)=-\left(2-m_{2}\right)$
$\Rightarrow \sqrt{3}+2 \sqrt{3} m_{2}+m_{2}=2$ or $\sqrt{3}+2 \sqrt{3} m_{2}-m_{2}=-2$
$\Rightarrow \sqrt{3}+(2 \sqrt{3}+1) m_{2}=2$ or $\sqrt{3}+(2 \sqrt{3}-1) m_{2}=-2$
$\Rightarrow m_{2}=\frac{2-\sqrt{3}}{(2 \sqrt{3}+1)}$ or $m_{2}=\frac{-(2+\sqrt{3})}{(2 \sqrt{3}-1)}$
Case I:
$m_{2}=\left(\frac{2-\sqrt{3}}{2 \sqrt{3}+1}\right)$
The equation of the line passing through point $(2,3)$ and having a slope of $\frac{(2-\sqrt{3})}{(2 \sqrt{3}+1)}$ is
$(y-3)=\frac{2-\sqrt{3}}{2 \sqrt{3}+1}(x-2)$
$(2 \sqrt{3}+1) y-3(2 \sqrt{3}+1)=(2-\sqrt{3}) x-2(2-\sqrt{3})$
$(\sqrt{3}-2) x+(2 \sqrt{3}+1) y=-4+2 \sqrt{3}+6 \sqrt{3}+3$
$(\sqrt{3}-2) x+(2 \sqrt{3}+1) y=-1+8 \sqrt{3}$
In this case, the equation of the other line is $(\sqrt{3}-2) x+(2 \sqrt{3}+1) y=-1+8 \sqrt{3}$.
Case II :
$m_{2}=\frac{-(2+\sqrt{3})}{(2 \sqrt{3}-1)}$
The equation of the line passing through point $(2,3)$ and having a slope of $\frac{-(2+\sqrt{3})}{(2 \sqrt{3}-1)}$ is
$(y-3)=\frac{-(2+\sqrt{3})}{(2 \sqrt{3}-1)}(x-2)$
$(2 \sqrt{3}-1) y-3(2 \sqrt{3}-1)=-(2+\sqrt{3}) x+2(2+\sqrt{3})$
$(2 \sqrt{3}-1) y+(2+\sqrt{3}) x=4+2 \sqrt{3}+6 \sqrt{3}-3$
$(2+\sqrt{3}) x+(2 \sqrt{3}-1) y=1+8 \sqrt{3}$
In this case, the equation of the other line is $(2+\sqrt{3}) x+(2 \sqrt{3}-1) y=1+8 \sqrt{3}$
Thus, the required equation of the other line is $(\sqrt{3}-2) x+(2 \sqrt{3}+1) y=-1+8 \sqrt{3}$ or $(2+\sqrt{3}) x+(2 \sqrt{3}-1) y=1+8 \sqrt{3}$.
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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,