Happy Chinese New Year Present Ms Wathall Coordinate Geometry C2

B) Distance between two points C) Equation of a circle after CNY Coordinate geometry A) Midpoint of a line B) Distance between two points C) Equation of a circle after CNY

Midpoints To find the midpoint between two points we find the average of the x’s and the average of the y’s. To find the midpoint between (x1,y1) and (x2,y2) we use x = x1+ x2 y = y1 +y2 2 2

Example 1 Find the midpoint between the points (-3,4) and (-1,6) Answer (-2,5)

Centre of a circle If we know that two points, A and B are a diameter of a circle then the centre of the circle is the midpoint of A and B! A B

Example 2 The line AB is a diameter of a circle where A and B are (-4,6) and (7,8). Find the coordinates of the centre. Answer (3/2, 7) Please complete ex 4A

Perpendicular lines Perpendicular lines look like this: Question plot the points E(-1,3) G(6,1) H(-3,-5) and F (5,-6) Join EF and GH and find their respective gradients Multiply these two gradients together. What do you notice?

Perpendicular Lines Two lines are perpendicular if there is a 900 angle between them. It looks like this : what is the relationship between their slopes? So m1 X m2 = -1 Or if two lines are perpendicular one gradient is the negative reciprocal of the other.

Example 3 Triangle ABC has vertices A(4, 5), B(8, 13) and C(-4, 9). Use the slopes to show that triangle ABC is a right angle.

A handy theorem The perpendicular from the centre of a circle to a chord bisects the chord

Example 4 3) Use y-y1 = m(x-x1) Ex 4B The line AB is a diameter of a circle centre C where A and B are (-2,5) and (2,9). The line l passes through C and is perpendicular to FG. Find the equation of l. Here draw a diagram! Follow these steps: 1) Find the midpoint of AB 2) Find the gradient of AB and take negative reciprocal 3) Use y-y1 = m(x-x1) Ex 4B l B A

Distance between two points Look at this Q(7,5) y 2 units 5 units P(2,3) x So using Pythagoras PQ2 = 52+22

Distance between two points Look at this Q(x2,y2) y (y2-y1) units P(x1,y1) (x2-x1)units x So using Pythagoras PQ2 = (x2-x1)2 + (y1-y2)2

Distance Formula The formula to find the distance between the two points (x1,y1) and (x2,y2) is: (we use Pythagoras!) PQ2 = (x1-x2)2 + (y1-y2)2 Taking the square root of both sides gives PQ = (x1-x2)2 + (y1-y2)2

Example 5 Find the distance between the two points (-4,3) and (2,-1) Answer 2√13 units

Another handy theorem Please complete Ex 4C The angle in a semi circle is 90 0 Please complete Ex 4C

Finding the centre Two perpendicular bisectors of two chords will intersect at the centre of the circle.

Equation of a circle centre (0,0) Let us look at a circle with the centre at (0,0) and whose radius is r. Using Pythagoras we have X2+y2 = r2 P (x,y)

Equation of a circle centre (a,b) We still use Pythagoras to find the equation of the circle with centre (a,b) Let us look at a numerical example first (x-2) 2 + (y-4) 2 = r 2 (x,y) (2,4)

General equation of a circle The general equation of the circle with centre (a,b) and radius r is (x-a) 2 + (y-b) 2 = r 2

Equation of a circle : centre (a, b) radius r y (x , y) r (a , b) x Equation of circle is (x – a)2 + (y – b)2 =r2

An applet Here is an applet to show the equation of circles with different centres

Example 1 Find the equation of the circle with centre (-4, 5) and radius 6

Example 2 Show that the circle (x-2) 2+ (y-5)2 = 13 passes through the point (4,8)

An extra example Find the centre and radius of the following circle X2 + y2 -10x -2y +17 = 0 Here use the method of completing the square X2 -10x +25 + y2 -2y +4 = -17+29 Can you complete this? Click here for lots of examples and a quick quiz

Another Example Find the centre and radius of the circles: x2 + y2 -2x +4y – 9 = 0

Finding points of intersection Find where the line y = x + 5 meets the circle x2 + (y- 2)2 = 29

Example Show the line y = x – 7 does not meet the circle (x + 2)2 + y2 = 33

More practice with CTS

Example 3 AB is a diameter of a circle where A is ( 4,7) and B is ( -8,3) . Find the equation of the circle. Step 1 : find the radius Step 2: Find the centre which is the midpoint of AB Step 3: use the general equation

An interesting circle theorem The angle between a tangent and the radius is 90 0 A tangent meets the circle at one point only Ex 4D

Example 4 Show that the line y = 2x-2 meets the circle (x-32)2 + (y-2) 2 = 20 at the points A and B. Show that AB is a diameter of the circle Step 1 solve simultaneously Step 2 find distance AB and prove it is 4√5 (why?) Ex 4E Mixed exercise 4F