Question

Find the equation of the circle passing through the points

(i) (0, 0), (5, 0) and (3, 3)

(ii) (1, 2), (3, – 4) and (5, – 6)

(iii) (20, 3), (19, 8) and (2, – 9)

Also, find the centre and radius in each case.

Answer 1

Before we start solving listed problems, students are advised to keep below information in mind.

The general equation of a circle is as follows:

x² + y² + 2gx + 2fy + c = 0

Where g, f and c are constants.

With

Centre: (-g, -f)

(i) (0, 0), (5, 0) and (3, 3)

The Circle equation is:

Let us apply Laplace Expansion to solve this problem:

On comparing above equation with the general form of circle, we get

2g = -5 ⇒ g = -2.5

2f = -1 ⇒ f = -0.5

c = 0

Now,

centre = (2.5, 0.5)

(ii) (1, 2), (3, – 4) and (5, – 6)

The Circle equation is:

Let us apply Laplace Expansion to solve this problem:

On comparing above equation with the general form of circle, we get

2g = -22 ⇒ g = -11

2f = -4 ⇒ f = -2

c = -25

Now,

Centre = (11, 2)

(iii) (20, 3), (19, 8) and (2, – 9)

The Circle equation is:

Let us apply Laplace Expansion to solve this problem:

On comparing above equation with the general form of circle, we get

2g = -14 ⇒ g = -7

2f = -6 ⇒ f = -3

c = -111

Now,

Centre = (7, 3)