In this case the statement above the diagram only has the equation of a cycle for us to notice, everything else is in the diagram and if not, you want to immediately include it so that you don’t forget the information in the middle of answering the question

4.1.1

• We are given the equation of a circle in a form that is not standard
• If we have it on standard form, we can identify the centre coordinates and length of radius
• Change the sign of a and b from the diagram to equation and vice versa, all the time
• If the number on the right of standard equation does not have a squared, the square root of that number is the radius
• And if it has a squared, that number itself is the radius
• Complete the square to get standard formula:

Add and subtract half the coefficient of x and y, and square it

But it is handier to simply add on both sides of the equation

Check the answer on the board bellow

4.1.2

• To find equation of a line we need: two coordinates or a gradient and a point
• In this case we can find the coordinates of A and the gradient of BA
• Construct triangle AFC as show in the board below
• C and F will be parallel to the x-axis and perpendicular to y-axis
• So, at F y=2 since C and F share the same y value on horizontal line FC
• AC is radius and is equal to 5 units, use Pythagoras to find AF
• Then we can find the coordinates of A
• To find gradient of tangent we will use gradient of radius and convert it using product of gradients rule

4.1.3

• Angle of inclination is an angle between x-axis and the line
• So, construct x-axis on A, of which is perpendicular to the y-axis
• Find the size of alpha as show in the board bellow

4.2

• Once again, we need to bring the equation to standard form by completing the square
• Calculate distance between centre of both circles and the sum of radiuses
• If distance is greater than sum of radius: circles do not touch
• If equal: the touch once
• If less: they touch twice

Answer: