Find the area of the triangle, the equations of whose sides are y = x, y = 2x and y – 3x = 4.
The given equations are
$y=x \ldots$ (i)
$y=2 x \ldots$ (ii)
and $y-3 x=4 \ldots$ (iii)
Let eq. (i), (ii) and (iii) represents the sides $A B, B C$ and $A C$ respectively of $\triangle A B C$
From eq. (i) and (ii), we get $x=0$ and $y=0$
Thus, $A B$ and $B C$ intersect at $(0,0)$
Solving eq. (ii) and (iii), we get
$\mathrm{y}=2 \mathrm{x} \ldots$ (ii)
and $y-3 x=4 \ldots$ (iii)
Putting the value of y = 2x in eq. (iii), we get
$2 x-3 x=4$
$\Rightarrow-x=4$
$\Rightarrow x=-4$
Putting the value of x = -4 in eq. (ii), we get
$y=2(-4)$
$\Rightarrow y=-8$
Thus, $B C$ and $A C$ intersect at $(-4,-8)$
Now, Solving eq. (iii) and (i), we get
$y-3 x=4 \ldots$ (iii)
and $y=x \ldots$ (i)
Putting the value of y = x in eq. (iii), we get
$x-3 x=4$
$\Rightarrow-2 x=4$
$\Rightarrow x=-2$
Putting the value of x = -2 in eq. (i), we get
$y=-2$
Thus, $A C$ and $A B$ intersect at $(-2,-2)$
So, vertices of triangle $\mathrm{ABC}$ are: $(0,0),(-4,-8)$ and $(-2,-2)$
$\therefore$ Area of $\Delta \mathrm{ABC}=\frac{1}{2}\left|\begin{array}{ccc}0 & 0 & 1 \\ -2 & -2 & 1 \\ -4 & -8 & 1\end{array}\right|$
$=\frac{1}{2}[0-0+1\{(-2)(-8)-(-2)(-4)\}]$
$=\frac{1}{2}[1\{16-8\}]$
$=\frac{1}{2}[8]$
$=4 \mathrm{sq} .$ units
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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,