Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes.
From the figure
AD = b units and AE = a units.
D(0, b), E(a, 0) and A(0, 0) lies on the circle. C is the centre.
The general equation of a circle: $(x-h)^{2}+(y-k)^{2}=r^{2}$
$\ldots(\mathrm{i})$, where $(\mathrm{h}, \mathrm{k})$ is the centre and $\mathrm{r}$ is the radius.
Putting $A(0,0)$ in (i)
$(0-h)^{2}+(0-k)^{2}=r^{2}$
$\Rightarrow \mathrm{h}^{2}+\mathrm{k}^{2}=\mathrm{r}^{2} \ldots$ (ii)
Similarly putting $D(0, b)$ in (i)
$(0-h)^{2}+(b-k)^{2}=r^{2}$
$\Rightarrow h^{2}+k^{2}+b^{2}-2 k b=r^{2}$
$\Rightarrow r^{2}+b^{2}-2 k b=r^{2}$
$\Rightarrow b^{2}-2 k b=0$
$\Rightarrow(b-2 k) b=0$
Either $b=$ Oork $=\frac{b}{2}$
Similarly putting $E(a, 0)$ in (i)
$(a-h)^{2}+(0-k)^{2}=r^{2}$
$\Rightarrow h^{2}+k^{2}+a^{2}-2 h a=r^{2}$
$\Rightarrow r^{2}+a^{2}-2 h a=r^{2}$
$\Rightarrow a^{2}-2 h a=0$
$\Rightarrow(a-2 h) a=0$
Either $a=$ Oorh $=\frac{a}{2}$
Centre $=C\left(\frac{\mathrm{a}}{2}, \frac{\mathrm{b}}{2}\right)$
$r^{2}=h^{2}+k^{2}$
$\Rightarrow \mathrm{r}^{2}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{4}$
Putting the value of $r^{2}, h$ and $k$ in equation (i)
$(x-h)^{2}+(y-k)^{2}=r^{2}$
$\left(x-\frac{a}{2}\right)^{2}+\left(y-\frac{b}{2}\right)^{2}=\frac{a^{2}+b^{2}}{4}$
$\Rightarrow x^{2}+y^{2}+\frac{a^{2}}{4}+\frac{b^{2}}{4}-x a-y b=\frac{a^{2}+b^{2}}{4}$
$\Rightarrow x^{2}+y^{2}-x a-y b=0$
which is the required equation.
- 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,
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,