Question

Wyznacz równanie okręgu opisanego na trójkącie ABC:

a) A(-9,0), B(-1,-4),C(3,4)

b) A(-1,1),B(3,-1), C(3,5)

Answer 1

x² + y² -2ax - 2by + c=0

 

a)

{(-9)² + 0² -2a*(-9) - 2b*0 + c=0

{(-1)² + (-4)² - 2a*(-1) -2b*(-4)+c=0

{3² + 4² -2a*3 - 2b*4 + c=0

 

{81 + 0 + 18a - 0 + c=0

{1 + 16 + 2a + 8b + c=0

{9 + 16 - 6a - 8b  + c = 0

 

{18a+c = -81

{2a+8b + c = -17

{-6a-8b + c = -25

 

{c=-81-18a

{2a+8b - 81-18a = -17

{-6a-8b-81-18a=-25

 

{c=-81-18a

{-16a+8b=64

{-24a-8b=56

 

{c=-81-18a

{-40a = 120   /:(-40)

 

{c=-81-18a

{a=-3

 

{c=-81-18*(-3)

{a=-3

 

{c=-27

{a=-3

{-16a+8b=64

 

-16*(-3) + 8b = 64

8b=16   /:8

b=2

 

{a=-3

{b=2

{c=-27

więc:

x² +y² +6x - 4y - 27=0

 

b)

{(-1)²+1² -2a*(-1) - 2b*1 + c=0

{3² + (-1)² - 2a*3 - 2b*(-1) + c=0

{3²+5² -2a*3-2b*5 + c=0

 

{1 + 1 + 2a -2b+c=0

{9 + 1 - 6a + 2b + c=0

{9+25 - 6a - 10b + c=0

 

{c=-2a+2b - 2

{-6a+2b - 2a+2b -2=-10

{-6a-10b -2a+2b-2=-34

 

{c=-2a+2b -2

{-8a+4b =-8     /:4

{-8a-8b=-32     /:8

 

{c=-2a+2b-2

{-2a+b = -2

{-a - b = -4

 

{c=-2a+2b-2

{-2a+b-a-b=-2-4

 

{c=-2a+2b-2

{-3a = -6    /:(-3)

 

{c=-2a+2b-2

{a=2

 

{c=-2*2 + 2b - 2

{-a-b=-4

{a=2

 

{c=2b-6

{-2-b=-4

{a=2

 

{c=2b-6

{b=2

{a=2

 

{c=-2

{b=2

{a=2

więc:

x² + y²-4x - 4y-2=0