Geo4.16

From Example Problems
Jump to: navigation, search

Find the equation to the circle passing through the points(a,0),(0,b),(a,b)\,


Let the equation of the circle be

x^{2}+y^{2}+2gx+2fy+c=0\,

The point (a,0) lies on the circle,then

a^{2}+0^{2}+2g(a)+2f(0)+c=0\,

2ga+c=-a^{2}\,

Let this equation be 1.

The point (0,b) lies on the circle,then

0^{2}+(b)^{2}+2g(0)+2f(b)+c=0\,

2fb+c=-b^{2}\,

Let this equation be 2.

The point (a,b) lies on the circle,then

(a)^{2}+(b)^{2}+2ga+2fb+c=0\,

2ga+2fb+c=-a^{2}-b^{2}\,

Let this equation be 3.

Solving 1 and 3 we get

2fb=-a^{2}+a^{2}-b^{2}\,

f={\frac  {-b}{2}}\,

Substituting this value in 2, we get

2\cdot {\frac  {-b}{2}}b+c=-b^{2}\,

c=0\,

Hence,

2ga=-a^{2}\,

g={\frac  {-a}{2}}\,

Now the equation of the circle is

x^{2}+y^{2}-ax-by=0\,


Main page:Geometry:Circles