Find the equation of the circle passing through the points (2, 3) and (–1, 1) and whose centre is on the line x – 3y – 11 = 0.
Let the equation of the required circle be $(x-h)^{2}+(y-k)^{2}=r^{2}$.
Since the circle passes through points (2, 3) and (–1, 1),
$(2-h)^{2}+(3-k)^{2}=r^{2} \ldots(1)$
$(-1-h)^{2}+(1-k)^{2}=r^{2} \ldots(2)$
Since the centre (h, k) of the circle lies on line x – 3y – 11 = 0,
h – 3k = 11 … (3)
From equations (1) and (2), we obtain
$(2-h)^{2}+(3-k)^{2}=(-1-h)^{2}+(1-k)^{2}$
$\Rightarrow 4-4 h+h^{2}+9-6 k+k^{2}=1+2 h+h^{2}+1-2 k+k^{2}$
$\Rightarrow 4-4 h+9-6 k=1+2 h+1-2 k$
$\Rightarrow 6 h+4 k=11 \ldots(4)$
On solving equations (3) and (4), we obtain $h=\frac{7}{2}$ and $k=\frac{-5}{2}$.
On substituting the values of $h$ and $k$ in equation (1), we obtain
$\left(2-\frac{7}{2}\right)^{2}+\left(3+\frac{5}{2}\right)^{2}=r^{2}$
$\Rightarrow\left(\frac{4-7}{2}\right)^{2}+\left(\frac{6+5}{2}\right)^{2}=r^{2}$
$\Rightarrow\left(\frac{-3}{2}\right)^{2}+\left(\frac{11}{2}\right)^{2}=r^{2}$
$\Rightarrow \frac{9}{4}+\frac{121}{4}=r^{2}$
$\Rightarrow \frac{130}{4}=r^{2}$
Thus, the equation of the required circle is
$\left(x-\frac{7}{2}\right)^{2}+\left(y+\frac{5}{2}\right)^{2}=\frac{130}{4}$
$\left(\frac{2 x-7}{2}\right)^{2}+\left(\frac{2 y+5}{2}\right)^{2}=\frac{130}{4}$
$4 x^{2}-28 x+49+4 y^{2}+20 y+25=130$
$4 x^{2}+4 y^{2}-28 x+20 y-56=0$
$4\left(x^{2}+y^{2}-7 x+5 y-14\right)=0$
$x^{2}+y^{2}-7 x+5 y-14=0$
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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,