Show that the points A(1, 0), B(2, - 7), c(8, 1) and D(9, - 6) all lie on the same circle. Find the equation of this circle, its centre and radius.
The general equation of a circle: (x - h)2 + (y - k)2 = r2
…(i), where (h, k) is the centre and r is the radius.
Putting (1, 0) in (i)
(1 - h)2 + (0 - k)2 = r2
h2 + k2 + 1 - 2h = r2 ..(ii)
Putting (2, - 7) in (i)
(2 - h)2 + ( - 7 - k)2 = r2
h2 + k2 + 53 - 4h + 14k = r2
(h2 + k2 + 1 - 2h) + 52 - 2h + 14k = r2
h - 7k - 26 = 0..(iii) [from (ii)]
Similarly putting (8, 1)
7h + k - 32 = 0..(iv)
Solving (iii)&(iv)
h = 5 and k = - 3
centre(5, - 3)
Radius = 25
To check if (9, - 6) lies on the circle, (9 - 5)2 + ( - 6 + 3)2 = 52
Hence, proved.
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