Find values of k if area of triangle is 4 square units and vertices are
(i) $(k, 0),(4,0),(0,2)$
(ii) $(-2,0),(0,4),(0, k)$
We know that the area of a triangle whose vertices are (x1, y1), (x2, y2), and
(x3, y3) is the absolute value of the determinant (Δ), where
$\Delta=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|$
It is given that the area of triangle is 4 square units.
$\therefore \Delta=\pm 4$
(i) The area of the triangle with vertices (k, 0), (4, 0), (0, 2) is given by the relation,
$\Delta=\frac{1}{2}\left|\begin{array}{lll}k & 0 & 1 \\ 4 & 0 & 1 \\ 0 & 2 & 1\end{array}\right|$
$=\frac{1}{2}[k(0-2)-0(4-0)+1(8-0)]$
$=\frac{1}{2}[-2 k+8]=-k+4$
$\therefore-K+4=\pm 4$
When $-k+4=-4, k=8$.
When $-k+4=4, k=0$
Hence, $k=0,8$.
(ii) The area of the triangle with vertices (−2, 0), (0, 4), (0, k) is given by the relation,
$\Delta=\frac{1}{2}\left|\begin{array}{ccc}-2 & 0 & 1 \\ 0 & 4 & 1 \\ 0 & k & 1\end{array}\right|$
$=\frac{1}{2}[-2(4-k)]$
$=k-4$
$\therefore k-4=\pm 4$
When $k-4=-4, k=0$.
When $k-4=4, k=8$.
Hence, $k=0,8$.
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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,