Find the equation of the circle through the points (1, 0), (-1, 0), an

1.	Find the equation of the circle through the points (1, 0), (-1, 0), and (0, 1).
Answer» Let the required circle be x² + y² + 2gx + 2fy + c = 0 The circle passes through (1, 0), (-1, 0) and (0, 1) (1, 0) ⇒ 1 + 0 + 2g(1) + 2f(0) + c = 0 2g + c = -1 … (1) (-1, 0) ⇒ 1 + 0 + 2g (-1) + 2f(0) + c = 0 -2g + c = -1 … (2) (0, 1) ⇒ 0 + 1 + 2g (0) + 2f(1) + c = 0 2f+ c = -1 ... (3) Now solving (1), (2) and (3). 2g + c = -1 … (1) -2g + c = -1 …(2) (1) + (2) ⇒ 2c = -2 ⇒ c = -1 Substituting c = -1 in (1) we get 2g – 1 = -1 2g = -1 + 1 = 0 ⇒ g = 0 Substituting c = -1 in (3) we get 2f – 1 =-1 ⇒ 2f = -1 + 1 = 0 ⇒ f = 0 So we get g = 0, f= 0 and c = -1 So the required circle will be x² + y² + 2(0) x + 2(0)y – 1 = 0 (i.e) x² + y² – 1 = 0 ⇒ x² + y² = 1

Find the equation of the circle through the points (1, 0), (-1, 0), and (0, 1).

Answer»

Let the required circle be

x² + y² + 2gx + 2fy + c = 0

The circle passes through

(1, 0), (-1, 0) and (0, 1)

(1, 0) ⇒ 1 + 0 + 2g(1) + 2f(0) + c = 0

2g + c = -1 … (1)

(-1, 0) ⇒ 1 + 0 + 2g (-1) + 2f(0) + c = 0

-2g + c = -1 … (2)

(0, 1) ⇒ 0 + 1 + 2g (0) + 2f(1) + c = 0

2f+ c = -1 ... (3)

Now solving (1), (2) and (3).

2g + c = -1 … (1)

-2g + c = -1 …(2)

(1) + (2) ⇒ 2c = -2 ⇒ c = -1

Substituting c = -1 in (1) we get

2g – 1 = -1

2g = -1 + 1 = 0 ⇒ g = 0

Substituting c = -1 in (3) we get

2f – 1 =-1 ⇒ 2f = -1 + 1 = 0 ⇒ f = 0

So we get g = 0, f= 0 and c = -1

So the required circle will be

x² + y² + 2(0) x + 2(0)y – 1 = 0

(i.e) x² + y² – 1 = 0 ⇒ x² + y² = 1