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**Scheme of work: Year 12 A-Level: Pure 1: Coordinate Geometry with Straight Lines and Circles**

- Find the gradient of a straight-line graph.
- Draw straight line graphs.
- Rearrange straight line equations into the form y = mx + c
- Find lengths and midpoints between two coordinate pairs.
- Complete the square of a quadratic function.
- Know and be able to apply Pythagoras Theorem
- Construct a circle in the form x
^{2}+ y^{2}= r^{2}. - Solve quadratic and linear equations simultaneously using the method of substitution
- A tangent is perpendicular to the radius of the circle at the point of intersection.

- Understand and use the equation of a straight line, including the forms y = mx + c and m(x – x
_{1}) = y – y_{1} - Gradient conditions for two straight lines to be parallel or perpendicular
- Be able to use straight line models in a variety of contexts
- Understand and use the coordinate geometry of the circle, including using the equation of a circle in the form (x – a)
^{2}+ (y -b)^{2}= r^{2} - Completing the square to find the centre and radius of a circle.
- Use of the following properties:
- the angle in a semicircle is a right angle
- the perpendicular from the centre to a chord bisects the chord
- the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point

- Points are collinear if they all lie on the same straight line.
- The equation of a straight line can be given as y = mx + c or ax + by + c = 0. Students need to be able to convert between the two forms.
- The equation a line in the form m(x – x
_{1}) = y – y_{1}can be derived from the gradient of a straight line where m = (y -y_{1}) / (x – x_{1}) - Parallel lines have the same gradient. Perpendicular lines have a negative reciprocal gradient.
- If two lines are perpendicular, the product of their gradients is -1.
- The equation of a circle with centre (a, b) can be given in the forms (x – a)
^{2}+ (y – b)^{2}= r^{2}and x^{2}+ y^{2}+ 2fx + 2gy + c = 0.

- Students can often become confused with the algebraic workings due to not drawing diagrams or diagrams lacking sufficient detail.
- A common mistake is to write the correct gradient of line in the form y = mx + c but write it incorrectly when converting the equation into the form ax + by + c = 0.
- The coordinates of the centre of a circle are sometimes given with the negatives. For instance, (x – 3)
^{2}+ (y – 4)^{2}= r^{2}can have the incorrect centre as (-3, -4). - Students can often become confused with the algebraic workings due to not drawing diagrams or diagrams lacking sufficient detail.

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