Question #cfe32

1 Answer
Apr 4, 2017

color(red)(7x^2+7y^2-67x+53y+216=0)

Explanation:

The general equation of a circle is given as x^2+y^2+2gx+2fy+c=0 where g, f, c are constants .

Substituting the given three points one-by-one in the above equation,

1. (6, -6)
36+36+12g-12f+c=0
implies 72+12g-12f+c=0

2.(3, -2)
9+4+6g-4f+c=0
implies 13+6g-4f+c=0

3.(7, -5)
49+25+14g-10f+c=0
implies 74+14g-10f+c=0

subtracting 2. from 3.;
74+14g-10f+c-(13+6g-4f+c)=0
implies 61+8g-6f=0 -------------------------( 4. )

subtracting 1. from 3.;
74+14g-10f+c-(72+12g-12f+c)=0
implies 2+2g+2f=0
implies 1+g+f=0 ----------------------( 5. )

from 5., g=-1-f ---------------------( 6. )

substituting this value of g in 4.;
61+8(-1-f)-6f=0
implies 61-8-8f-6f=0
implies 53=14f
implies color(red)(f=53/14)

substituting this value of f in 6.;
g=-1-53/14 = (-14-53)/14

implies color(red)(g=-67/14)

substituting these values of f and g in any of the equations 1., 2., 3., to obtain the value of c.

Let's use 2.
13-6*67/14-4*53/14+c=0
implies -216/7+c=0
implies color(red)(c=216/7)

substituting these values of g, f, c in the general equation of a circle [x^2+y^2+2gx+2fy+c=0]

x^2+y^2-67/7x+53/7y+216/7=0

implies color(red)(7x^2+7y^2-67x+53y+216=0)

is the required equation of the circle.