THE CIRCLE: DEFINITION , GENERAL EQUATION, EQUATION OF TANGENT TO CIRCLE AND LENGTH OF THE TANGENT

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.

From (x – a)² + (y – b)² = r²

r² – 2ax + a² + y² – 2by + b² – r² = 0 x² + y² – 2ax – 2by + a² + b² – r² = 0

The above equation can be written as x² + y² + 2gx + 2fy + c = 0 Where a = – g, b = -f, c = – a² + b² – r²

Hence: x² + y² + 2gx + 2fy + c = 0 is called the general equation of a circle. observe the following about the general equation

1. It is a second degree equation in x and y iiThe co-efficient of x² and y² 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 = r²

(x-3)² + (y+2)² = 4²

x² – 6x + 9 + y² + 4y + 4 = 16

x² + y² – 6x + 4y + 9 + 4 – 16 = 0 x² + y² – 6x + 4y – 3 = 0

Find the centre and radius of a circle whose equation is x² + y² – 6x + 4y – 3 = 0

Solution:

x² + y² – 6x + 4y – 3 = 0 x² – 6x + y² + 4y = + 3

Complete the square for x and y x² – 6x + 9 + y² 4y + 4 = 3 + 9 + 4

(x – 3)² + (y + 2)² = 16

Compare with (x – a)² + (y – b) = r²

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 x² + y² + 2gx + 2fy + c = 0 2² + 3² + 4g + 6f + c = 0

5² + 4² + 10g + 8f + c = 0

3² + 7² + 6f + 14f + c = 0

Simplify the 3 equations f = – 107 / 22

g = – 67 / 22

c = – 312 / 11

From (x – a)² + (y – b)² = r²

r² – 2ax + a² + y² – 2by + b² – r² = 0 x² + y² – 2ax – 2by + a² + b² – r² = 0

The above equation can be written as x² + y² + 2gx + 2fy + c = 0 Where a = – g, b = -f, c = – a² + b² – r²

Hence: x² + y² + 2gx + 2fy + c = 0 is called the general equation of a circle. observe the following about the general equation

1. It is a second degree equation in x and y iiThe co-efficient of x² and y² 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 = r²

(x-3)² + (y+2)² = 4²

x² – 6x + 9 + y² + 4y + 4 = 16

x² + y² – 6x + 4y + 9 + 4 – 16 = 0 x² + y² – 6x + 4y – 3 = 0

Find the centre and radius of a circle whose equation is x² + y² – 6x + 4y – 3 = 0

Solution:

x² + y² – 6x + 4y – 3 = 0 x² – 6x + y² + 4y = + 3

Complete the square for x and y x² – 6x + 9 + y² 4y + 4 = 3 + 9 + 4

(x – 3)² + (y + 2)² = 16

Compare with (x – a)² + (y – b) = r²

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 x² + y² + 2gx + 2fy + c = 0 2² + 3² + 4g + 6f + c = 0

5² + 4² + 10g + 8f + c = 0

3² + 7² + 6f + 14f + c = 0

Simplify the 3 equations f = – 107 / 22

g = – 67 / 22

c = – 312 / 11

EQUATION OF TANGENT TO A CIRCLE AT POINT (x1, y1)

Evaluation

Find the equation of the tangent to the circle 1.           x² + y² + 4x – 10y – 12 = 0        at (3,1)

2.        x² + y² – 6x – 3y = 16                 at (-2,0)

GENERAL/REVISION EVALUATION

1. Find the equation of the circle with center (1,3) and radius √5
2. Find the equation of the circle that passed through the point (0,0), (2,0) and (3, -1)
3. Find the equation of the circumcircle of the triangle whose vertices are A( 2,3) B ( 5,4) and C(3,7)
4. Find the length of the tangent to the circle x² + y² -2x -4y -4 =0 from the point ( 8, 10)

READING ASSIGNMENT

Read equation of a circle, Further Mathematics Project II, page 205 – 210

WEEKEND ASSIGNMENT

1. What is the radius of the circle whose equation is x² + y² – 6x – 7=0 (a) 2 (b) 3 (c) 4 (d) 9
2. Which of the following is not an equation of a circle? (a) x² + y² =4 (b) x² + y² – 2x – 3=0 (c) x² + y² – 2xy + 4x – 6y + 1 = 0 (d) 2x² + 2y² – 6x + 4y + 3 = 0
3. The equation of a circle with centre (-2, 5) and radius 3 units is (a) x² + y² + 4x – 10y + 20 = 0 (b) x² + y² + 4x – 10y + 26 = 0 (c) x² + y² + 4x – 10y – 38 = 0 (d) x² + y² + 4x – 10y + 39 = 0
4. Find the coordinates of the centre of the circle 2x² + 2y² – 4x + 12y – 7 = 0 is (a) (-1, 3) (b) 1, 3) (c) (2, -6) (d) 41 (e) 10
5. The equation of a circle of radius 3 is x² + y² + 10x – 8y + k = 0. Find the value of the constant K (a) -50 (b) 18 (c) 32 (d) 41 (e) 10

THEORY

1. The equation of a circle is x² + y² – 10x + 8y = 0 find (i) its radius (ii) its area
2. A circle passes through the points (0,3) and (4,1), if the centre of the circle is on the x – axis, find the equation of the circle.