Back to: Further Mathematics SS2

Definition:

A circle is defines as the locus of point equidistant from a fixed point. A circle is completely specified by the centre and the radius.

Equation of a circle with centre (a,b) and radius r.

From (x – a)^{2} + (y – b)^{2} = r^{2}

r^{2} – 2ax + a^{2} + y^{2} – 2by + b^{2} – r^{2} = 0

x^{2} + y^{2} – 2ax – 2by + a^{2} + b^{2} – r^{2} = 0

The above equation can be written as x^{2} + y^{2} + 2gx + 2fy + c = 0

Where a = – g, b = -f, c = – a^{2} + b^{2} – r^{2}

Hence: x^{2} + y^{2} + 2gx + 2fy + c = 0 is called the general equation of a circle. observe the following about the general equation

i. It is a second degree equation in x and y

iiThe co-efficient of x^{2} and y^{2} are equal

iiiIt has no xy term

*Examples:*

1.Find the equation of a circle of centre (3, -2) radius 4 unit

*Solution:*

a = 3, b = -2 and r = 2

(x-a)^{2} + (y-b)^{2} = r^{2}

(x-3)^{2} + (y+2)^{2} = 4^{2}

x^{2} – 6x + 9 + y^{2} + 4y + 4 = 16

x^{2} + y^{2} – 6x + 4y + 9 + 4 – 16 = 0

x^{2} + y^{2} – 6x + 4y – 3 = 0

Find the centre and radius of a circle whose equation is x^{2} + y^{2} – 6x + 4y – 3 = 0

*Solution:*

x^{2} + y^{2} – 6x + 4y – 3 = 0

x^{2} – 6x + y^{2} + 4y = + 3

Complete the square for x and y

x^{2} – 6x + 9 + y^{2} 4y + 4 = 3 + 9 + 4

(x – 3)^{2} + (y + 2)^{2} = 16

Compare with (x – a)^{2} + (y – b) = r^{2}

Equation of a circle passing through 3 points

Find the equation of the circumcircle of the triangle whose vertices are A (2,3) B (5,4) and C (3,7)

Solution:

The equation of the circle x^{2} + y^{2} + 2gx + 2fy + c = 0

2^{2} + 3^{2} + 4g + 6f + c = 0

5^{2} + 4^{2} + 10g + 8f + c = 0

3^{2} + 7^{2} + 6f + 14f + c = 0

Simplify the 3 equations

f = – 107 / 22

g = – 67 / 22

c = – 312 / 11

Hence, the equation of the circle is

x^{2} + y^{2} + 2 x + 2 y + = 0

11x^{2} + 11y^{2} – 67x – 107y + 312 = 0

a = 3, b = -2, r^{2} = 16,

r = = 4

hence the centre is (3, -2) and the radius is 4 unit

**Evaluation:**

1.Find the equation of the circle (-1 -1) and radius 3

Find the centre and radius of the circle x^{2} + y^{2} – 6x + 14y + 49 = 0

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