Find the radius and centre of a circle passing through (7, 1), (0, 0), (_1, 7).

Answer 1

circle through origin and (7,1) & (1,7)

x^2 +y^2 +ax+by = c, plug in the points

(0,0) gives c = 0

x^2 +y^2 +ax +by = 0

(1,7) gives 1+49 +a +7b = 0

a +7b = -50

(7,1) gives 49+1 +7a +b = 0

-50 = 7a +b

7a +49b = -350

7a +b = -50 subtract

48b = -300

b = -75/12 =-25/4

a = -50-7(-25/4) =-200/4+175/4=-25/4

x^2+y^2 -6.25x - 6.25y = 0

complete the squares

(x-25/8)^2 +(y-25/8)^2 = 2(25/8)^2

circle center = (25/8,25/8)

with radius squared = 2(25/8)^2,

radius = (25/8)sqr2

Answer 2

Let (a, b) is center

Then a² + b² = (a - 7)² + (b - 1)² = (a - 1)² + (b - 7)² = r²

14a + 2b = 50; a + 7b = 25

2a + 14b = 50; 7a + b = 25

a = b = 25/8; r² = 625/64 + 625/64 = 625/32

(x - 25/8)² + (y - 25/8)² = 625/32