Question

Find the equation of the circle. 

(a) Passing through the origin, having its centre on the x- axis and radius 2 units. 

(b) Passing through (2, 3) having its centre on the x-axis and radius 5 units. 

(c) Passing through the points (5, 1), (3, 4) and has its centre on the x – axis. 

(d) Passing through the points (1, 2) and (2, 1) and has its centre on the y – axis.

Answer 1

(a) Let A = (0, 0), centre on x – axis ⇒ C = (-g, 0), r = 2 units 

Required equation is x² + y² + 2gx + 2fy + c = 0 

It passes their (0,0) ⇒ c = 0, r = \(\sqrt{g^2 + f^2 - c}\)

2 = \(\sqrt{g^2 + 0 - 0}\) ⇒ g = ±2 

Equation of the circle is x² + y² ± 4x = 0. 

(b) Let the general equation of the circle be x² + y² + 2gx + 2fy + c = 0. 

It passes through (2,3) ⇒ (2)² + (3)² + 2g(2) + 2f(3) + c = 0 

4g + 6f + c + 13 = 0 ….(1) 

Centre (-g, -f) lies on x – axis ⇒ f = 0 …(2) 

r = \(\sqrt{g^2 + f^2 - c}\) = 5 = \(\sqrt{g^2 - c}\) = 25 = g² – C. 

⇒ g² – 25 = c. ….(3) 

Solving 1, 2 & 3 we get g² – 25 + 4g + 13 = 0 

g² + 4g – 12 = 0 

⇒ (g + 6) (g – 2) = 0 ⇒ g = -6 or 2 

When g = -6, c = (-6)² – 25 = 36 – 25 = 11 

g = 2, c = (2)² – 25 = 4 – 25 = -21 . 

∴ The required circles are x² + y² – 12x + 11 = 0 

x² + y² + 4x – 21 = 0. 

(C) Let the required equation of the circle is x² + y² + 2gx + 2fy + c = 0 

It passes through (5, 1) & (3, 4) (5, 1) (5)² + (1)² + 2g(5) + 2f (1) +C = 0 

10g + 2f + C + 26 = 0 …..(1) 

(3,4)3² + 4² + 2g (3) + 2f (4) + c= 0 

68 + 8f + C + 25 = 0 …..(2) 

& the centre (- g, – f) lies on x-axis ⇒ f = 0 ….(3) 

Solving 1 & 2 we get

⇒ 2x² + 2y² – x – 47 = 0. 

(d) Let required equation of the circle is x² + y² + 2gx + 2fy + c = 0 

It passes through the points (1,2) & (2, 1) & centre lies on y-axis ⇒ g = 0 ….(1) 

(1,2)1² + 2² + 2g(1) + 2f(2) + c = 0 ⇒ 4f + C + 5 = 0 ….(2) 

(2, 1)2² + 1² + 2g(2) + 2f(1) + c = 0 ⇒ 2f+c + 5 = 0 ….(3) 

Equation (3) – eqn. (2) gives 2f = 0 ⇒ f = 0 

g = 0, f = 0 ⇒ c= -5 

∴ Required equation of the circle is x² + y² – 5 = 0