Class 11

Math

3D Geometry

Conic Sections

Find the equation of the circle passing through the points $(2,3)$ and $(−1,1)$ and whose centre is on the line $x−3y−11=0$

$x_{2}+y_{2}+2gx+2fy+c=0$ .....(1)

where $(−g,−f)$ is center and $g_{2}+f_{2}−c$ is the radius.

Since, the circle passes through $(2,3)$, so it satisfies eqn (1)

$⇒4+9+4g+6f+c=0$

$⇒4g+6f+c=−13$ .....(2)

Since, the circle passes through $(−1,1)$, so it satisfies eqn (1)

$⇒1+1−2g+2f+c=0$

$⇒−2g+2f+c=−2$ .....(3)

Subtracting eqn (3) from eqn (2), we get

$6g+4f=−11$ ....(4)

Given center lies on the line $x−3y−11=0$

Since, center $(−g,−f)$ satisfies this equation.

$⇒−g+3f=11$ ....(5)

Solving eqn (4) and (5), we get

$g=2−7 ,f=25 $

Put this value in (2), we get

$c=−14$

Substituting these values in (1),

$x_{2}+y_{2}−7x+5y−14=0$

which is the equation of required circle.