Co-ordinate Geometry of the Circle

Show the format of the 3 different types of equations of a circle:

Show the different ways of finding the centre and r for each equation:

x² +y² = number

• centre is (0,0), r = √number

(x -2)² +(y +4)² = number

• centre is (2,-4), r = √number

x² +y² + 10x -4y + 13 = 0

• use formula in log tables

• let 2g = 10, the cooefficient of x

• let 2f = -4, the coefficient of y

• solve for g and f

• insert into centre formula

• insert into radius formula

What significance do the powers have in the equation of a circle?

shows that the line crosses the x-axis twice

Criteria of an equation for a circle:

the power of x and y must both be 2

they must have the same coefficient

there is no xy term in the equations

What is the condition for using the centre and radius formula?

the coefficient of both x and y are 1

How do you find where a circle crosses the y axis?

let y = 0

form a quadratic, solve

let x = 0

form a quadratic, solve

When given the coordinates of the end points of the diameter of the circle

(4,-2) and (8,6)

how do you find the radius?

use the midpoint formula

(6,2) is the centre

use distance formula from the centre to an end point

How do you find the equation of a circle when given the centre and the radius?

x = 6 and y = 2 are roots, (x -6) and (y -2) are factor

let equation = radius², use formula log tables

(x -6)² + (y -2)² = 20

How do you find the intersection between a circle and a straight line?

rewrite the equation for the line with x or y as the subject

sub that into the equation for the circle

form a quadratic

solve for y

sub in y, solve for x

What is the relationship between the radius and a tangent?

they are perpendicular

What is a tangent?

it is a line that kisses or touches a circle at one point

is when the perpendicular distance of the line from the centre of a circle is the same length as the radius

How do you find the equation of a parallel tangent when given the point it touches the circle

(5,4)

and the equation of the circle

x² +y² = 41

find the slope of the radius, between centre and point

find perpendicular slope

insert information for the formula of a straight line

the parallel tangent has the same slope, translate across to find the point

(-5,-4)

How do you show that a line and a circle do not intersect?

find radius

find the perpendicular distance between the line and the circle

show that the perpendicular distance is greater than the radius

How do you show that a line is a tangent to a circle?

show that the perpendicular distance = to the radius

How do you show that two circles touch externally?

find the centre and radius of both circles

find the distance between the two centres

find the sum of the radi

confirm that the distance = sum of the radi