A variable line passes through a fixed point P. The algebraic sum of the perpendiculars drawn from the points (2, 0), (0, 2) and (1, 1) on the line is zero. Find the coordinates of the point P.
Let the variable line be $a x+b y=1$
We know that, length of the perpendicular from $(p, q)$ to the line $a x+b y+c=0$
is
$\mathrm{d}=\left|\frac{\mathrm{ap}+\mathrm{bq}+\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}\right|$
$\mathrm{d}_{1}=\left|\frac{2 \times \mathrm{a}+0 \times \mathrm{b}-1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}\right|$
$=\frac{2 a-1}{\sqrt{a^{2}+b^{2}}}$
Now, perpendicular distance from B $(0,2)$
$\mathrm{d}_{2}=\left|\frac{0 \times \mathrm{a}+2 \times \mathrm{b}-1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}\right|$
$=\frac{2 b-1}{\sqrt{a^{2}+b^{2}}}$
Now, perpendicular distance from $C(1,1)$
$\mathrm{d}_{3}=\left|\frac{1 \times \mathrm{a}+1 \times \mathrm{b}-1}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}\right|$
$=\frac{a+b-1}{\sqrt{a^{2}+b^{2}}}$
It is given that the algebraic sum of the perpendicular from the given points $(2,$, $0),(0,2)$ and $(1,1)$ to this line is zero.
$d_{1}+d_{2}+d_{3}=0$
Substituting the values we get
$\therefore \frac{2 a-1}{\sqrt{a^{2}+b^{2}}}+\frac{2 b-1}{\sqrt{a^{2}+b^{2}}}+\frac{a+b-1}{\sqrt{a^{2}+b^{2}}}=0$
⇒ 2a – 1 + 2b – 1 + a + b – 1 = 0
⇒ 3a + 3b – 3 = 0
⇒ a + b – 1 = 0
⇒ a + b = 1
So, the equation ax + by = 1 represents a family of straight lines passing through a fixed point.
Comparing the equation ax + by = 1 and a + b = 1, we get
x = 1 and y = 1
So, the coordinates of fixed point is (1, 1)
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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
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