### Completing the Square

Completing the square is a technique required to find equations of circle

Example 1

Example 1

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Example 1

A typical coordinate geometry question from the O Levels

Given that AC = 3AB, find the coordinates of point B.

Remarks:

A lot of students will try to use the distance formula to form simultaneous equations in x and y to find the coordinates of B. This is a very long and tedious method. A much simpler method is to use ratio and proportion to find the coordinates of B

To find the x coordinate of B

We can use the same method to find the y coordinate of point B

Therefore the coordinates of point B is (4,4)

A typical coordinate geometry question from the O Levels

Given that AC = 3AB, find the coordinates of point B.

Remarks:

A lot of students will try to use the distance formula to form simultaneous equations in x and y to find the coordinates of B. This is a very long and tedious method. A much simpler method is to use ratio and proportion to find the coordinates of B

To find the x coordinate of B

We can use the same method to find the y coordinate of point B

Therefore the coordinates of point B is (4,4)

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Technique 1: Finding the gradient of a line

Parallel lines Parallel lines have the same gradient

Perpendicular lines

If a line has a gradient of 1/2 , the gradient of the line perpendicular to it is -2.

-2 is the negative reciprocal of 1/2. In other words, we "flip" the fraction 1/2 to get 2 and attach a negative sign

Technique 2: Finding equation of a straight line

The general format of a straight line is y = mx + c

where m is the gradient and,

c is the y intercept

Example 2

Find the equation of a line that contains the points (1,2) and (3,4).

From the example above we know that gradient of the line is 1.

Hence the equation of the line is: y = 1x + c

To find the y intercept, c, we sub in one of the points

I choose the point (1,2) and sub x =1, y = 2

2 = 1 + c

Hence c = 1

The equation of the line is y = x + 1

Let us combine the techniques we have learnt so far to solve a typical O Level question

Question 1

The figure above shows a trapezium where AB is parallel to CD. Both the lines AB and CD…

Parallel lines Parallel lines have the same gradient

Perpendicular lines

If a line has a gradient of 1/2 , the gradient of the line perpendicular to it is -2.

-2 is the negative reciprocal of 1/2. In other words, we "flip" the fraction 1/2 to get 2 and attach a negative sign

Technique 2: Finding equation of a straight line

The general format of a straight line is y = mx + c

where m is the gradient and,

c is the y intercept

Example 2

Find the equation of a line that contains the points (1,2) and (3,4).

From the example above we know that gradient of the line is 1.

Hence the equation of the line is: y = 1x + c

To find the y intercept, c, we sub in one of the points

I choose the point (1,2) and sub x =1, y = 2

2 = 1 + c

Hence c = 1

The equation of the line is y = x + 1

Let us combine the techniques we have learnt so far to solve a typical O Level question

Question 1

The figure above shows a trapezium where AB is parallel to CD. Both the lines AB and CD…