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COORDINATE GEOMETRY

  • Sum based on Finding Radius of a circle

2 of 3

C

(a, b)

A

B

(3, -3)

(-1, 1)

(3, 5)

O

Q. Find the co-ordinates of the centre of a circle passing through the

points A (3, 5), B (–1, 1) and C (3, –3). Also, find the radius of the circle.

Sol.

Let, O

(a, b)

OA =

OB =

OC

[radii of the same circle]

OA

= OB

+

–

–

(

)

(

)

a

3

b

5

2

2

+

–

–

[

]

(

)

a

(–1)

b

1

2

2

=

(a – 3)2

+

(b – 5)2

∴

=

(a + 1)2

+

(b – 1)2

–

6a

+

9

+

b2

–

10b

+

25

=

a2

+

2a

+

1

+

b2

–

2b

+

1

a2

∴

∴

– 6a

10b

34

–

+

=

2a

–

2b

+

2

∴

– 6a

2a

2b

–

+

=

10b

–

2

–

34

∴

–8a

8b

–

=

–32

…(i)

∴

a

b

+

=

4

Let the co-ordinates of O be (a, b).

Let us join OA, OB and OC.

What can we say about OA, OB and OC?

OA = OB = OC

(Radii of the same circle)

Squaring on both the sides.

Collecting variables on one side and constants on other side.

Which is this algebraic identity?

(a + b)2

What is the expansion ?

a2 + 2ab + b2

x1 = 3,

y1 = 5

x2 = a,

y2 = b

x1 = –1,

y1 = 1

x2 = a,

y2 = b

What is the expansion ?

a2 − 2ab + b2

Which is this algebraic identity?

(a − b)2

Which is the formula to find length of OA and OB?

+

–

–

(

)

(

)

x2

x1

y2

y1

2

2

Let the co-ordinates of O be (x2, y2)

Let the co-ordinates of A be (x1, y1)

Let the co-ordinates of O be (x2, y2)

Let the co-ordinates of B be (x1, y1)

3 of 3

Q. Find the co-ordinates of the centre of a circle passing through the

points A (3, 5), B (–1, 1) and C (3, –3). Also, find the radius of the circle.

Sol.

Let, O

(a, b)

OA =

OB =

OC

(radii of the same circle)

OB

= OC

C

(a, b)

A

B

(3, -3)

(-1, 1)

(3, 5)

O

+

–

–

[

]

(a

3)

b

(–3)

2

2

=

+

–

–

[

]

(

)

a

(–1)

b

1

2

2

∴ (a + 1)2

+

(b – 1)2

∴

=

(a – 3)2

+

(b + 3)2

+

2a

+

1

+

b2

–

2b

+

1

=

a2

–

6a

+

9

+

b2

+

6b

+

9

a2

∴

2a

2b

2

–

+

=

–6a

+

6b

+

18

∴

2a

6a

6b

+

–

=

2b

–

18

–

2

∴

8a

8b

–

=

16

…(ii)

∴

a

b

–

=

2

Collecting variables on one side and constants on other side.

Squaring on both the sides.

a + b = 4 …(i)

To get the values of two variables, we require two linear equations.

x1 = –1,

y1 = 1

x2 = a,

y2 = b

x2 = a,

y2 = b

x1 = 3,

y1 = –3

Which is the formula to find length of OB and OC?

+

–

–

(

)

(

)

x2

x1

y2

y1

2

2

Let the co-ordinates of O be (x2, y2)

Let the co-ordinates of B be (x1, y1)

Let the co-ordinates of O be (x2, y2)

Let the co-ordinates of C be (x1, y1)