Equation of a Circle

    OCR
    GCSE

    The Cartesian equation of a circle, (x-a)² + (y-b)² = r², represents the fundamental locus of points at a fixed distance from a center (a,b). Candidates must demonstrate fluency in converting between the expanded general form x² + y² + 2gx + 2fy + c = 0 and the standard form via completing the square. Assessment focuses on the interplay between algebraic manipulation and geometric reasoning, specifically regarding the derivation of tangent and normal equations and the analysis of intersection points with linear loci.

    0
    Objectives
    3
    Exam Tips
    4
    Pitfalls
    4
    Key Terms
    5
    Mark Points

    What You Need to Demonstrate

    Key skills and knowledge for this topic

    • Award B1 for correctly stating the centre (a, b) and radius r directly from the standard form (x-a)² + (y-b)² = r²
    • Award M1 for a complete method to find intersection points by substituting a linear equation y = mx + c into the circle equation
    • Award M1 for using the perpendicular gradient property (m₁ × m₂ = -1) to determine the gradient of the tangent from the radius
    • Award M1 for attempting to complete the square on both x and y terms to identify the centre and radius from the general expanded form
    • Award A1 for the fully correct equation of the tangent, often required in the form ax + by + c = 0 with integer coefficients

    Example Examiner Feedback

    Real feedback patterns examiners use when marking

    • "You correctly identified the centre, but check your radius calculation — did you remember to square root the constant?"
    • "Your method for finding the tangent gradient is correct; ensure you use the negative reciprocal of the radius gradient"
    • "To secure the marks for 'Show that', you must explicitly state the condition b² - 4ac = 0 at the end of your working"
    • "Be careful with signs when completing the square — remember that (x-a)² expands to x² - 2ax + a²"

    Marking Points

    Key points examiners look for in your answers

    • Award B1 for correctly stating the centre (a, b) and radius r directly from the standard form (x-a)² + (y-b)² = r²
    • Award M1 for a complete method to find intersection points by substituting a linear equation y = mx + c into the circle equation
    • Award M1 for using the perpendicular gradient property (m₁ × m₂ = -1) to determine the gradient of the tangent from the radius
    • Award M1 for attempting to complete the square on both x and y terms to identify the centre and radius from the general expanded form
    • Award A1 for the fully correct equation of the tangent, often required in the form ax + by + c = 0 with integer coefficients

    Examiner Tips

    Expert advice for maximising your marks

    • 💡When asked to show a line is a tangent, explicitly calculate the discriminant of the resulting quadratic and state 'Since b²-4ac=0, there is one repeated root, so the line is a tangent'
    • 💡Always sketch a diagram for coordinate geometry problems involving circles; visualizing the centre, radius, and tangent points often reveals simpler geometric paths to the solution
    • 💡Memorize the circle theorems from GCSE Mathematics, as OCR Further Maths often requires applying 'angle in a semicircle is 90 degrees' to find equations of circles given diameter endpoints

    Common Mistakes

    Pitfalls to avoid in your exam answers

    • Stating the radius as the value on the RHS rather than its square root (e.g., for = 16, stating r=16 instead of r=4)
    • Sign errors when extracting coordinates of the centre, specifically interpreting (x+3)² as a centre coordinate of x=3 rather than x=-3
    • Algebraic errors when completing the square, particularly failing to subtract the square of the half-coefficient from the constant term side
    • Assuming the gradient of the tangent is the same as the radius, rather than the negative reciprocal

    Study Guide Available

    Comprehensive revision notes & examples

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    Key Terminology

    Essential terms to know

    Standard and general forms of the circle equation
    Coordinate geometry of tangents and normals
    Intersection of lines and circles (discriminant analysis)
    Geometric properties of chords and bisectors

    Likely Command Words

    How questions on this topic are typically asked

    Find
    Show that
    Determine
    Calculate
    Prove

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