Coordinate Geometry with Straight Lines and Circles

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

Prerequisite Knowledge

• 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 x2 + y2 = r2.
• 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.

Success Criteria

• Understand and use the equation of a straight line, including the forms y = mx + c and m(x – x1) = y – y1
• 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 = r2
• 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

Key Concepts

• 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 – x1) = y – y1 can be derived from the gradient of a straight line where m = (y -y1) / (x – x1)
• 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 = r2 and x2 + y2 + 2fx + 2gy + c = 0.

Common Misconceptions

• 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 = r2 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.

Coordinate Geometry with Straight Lines and Circles Resources

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