Show that the points A(1, 0), B(2, – 7), C(8, 1) and D(9, – 6) all

1.	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.
Answer» 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. Consider points (1, 0), (2, – 7) and (8, 1) lie on the circle. Putting (1, 0), (2, – 7) and (8, 1) in (i) Putting (1, 0) ⇒ h² + k² + 1 – 2h = r² ……(ii) Putting (2, – 7) ⇒ h² + k² + 53 – 4h + 14k = r² …….(iii) Putting (8, 1) ⇒ (8 – h)² + (1 – k)² = r² h² + k² + 65 – 16h – 2k = r²………….(iv) Subtract (ii) from (iii), we get h – 7k – 26 = 0 ……(v) Subtract (ii) from (iv), we get 7h + k – 32 = 0 ……(vi) Solving (v) and (vi) h = 5 and k = – 3 Equation (iv) ⇒ r = 25 [using h = 5 and k = – 3] Therefore, Centre (5, – 3) Radius = 25 Check for (9, – 6): To check if (9, – 6) lies on the circle, (9 – 5)² + ( – 6 + 3)² = 5² 25 = 25 Which is true. Hence, all the points are lie on circle.

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.

Answer»

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.

Consider points (1, 0), (2, – 7) and (8, 1) lie on the circle.

Putting (1, 0), (2, – 7) and (8, 1) in (i)

Putting (1, 0) ⇒ h² + k² + 1 – 2h = r² ……(ii)

Putting (2, – 7) ⇒ h² + k² + 53 – 4h + 14k = r² …….(iii)

Putting (8, 1) ⇒ (8 – h)² + (1 – k)² = r²

h² + k² + 65 – 16h – 2k = r²………….(iv)

Subtract (ii) from (iii), we get

h – 7k – 26 = 0 ……(v)

Subtract (ii) from (iv), we get

7h + k – 32 = 0 ……(vi)

Solving (v) and (vi)

h = 5 and k = – 3

Equation (iv) ⇒ r = 25

[using h = 5 and k = – 3]

Therefore,

Centre (5, – 3)

Radius = 25

Check for (9, – 6):

To check if (9, – 6) lies on the circle,

(9 – 5)² + ( – 6 + 3)² = 5²

25 = 25

Which is true.

Hence, all the points are lie on circle.