Answer to Question #1347 in Analytic Geometry for jun- jun villanueva

Question

Find the equation of the circle inscribed the triangle determind by the lines:2x-3y+21=0;3x-2y-6=0 and 2x+3y+9=0.

Answer 1

Firstly, we have to find the points of intersection of thelines:
1. 2x-3y+21=0 and 3x-2y-6=0 :
(12, 15)

2. 2x-3y+21=0 and 2x+3y+9=0 :
(-7.5, 2)
3. 3x-2y-6=0 and 2x+3y+9=0 :
(0; -3).
As the equation of the circle is
R² = (x-a)²+ (y-b)²,
we have the final system of equations for a and b:
(12-a)² + (15-b)² = (-7.5-a)²+ (2 – b)²
(12-a)² + (15-b)² =a² + (-3-b)²
144 + 225 – 24a -30b = 56.25 + 4 + 15a – 4b;
144+ 225 – 24a -30b = 9 – 6b.
308.75 = 39a + 26b;
360 = 24a +24b.
(-6.25, 21.25)
Thus R² = (12+ 6.25)² + (15-21.25)²= 372.125
Answer: (x+6.25)² + (y-21.25)²=372.125