AQA A-Level Mathematics: Circle Coordinate Geometry — mark scheme explained

The question

Find the equation of a circle with centre (3, -4) and radius 5. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  The standard form of the equation of a circle is (x - a) 2 + (y - b) 2 = r 2 .

• S2

  Here, the centre (a, b) is (3, -4) and the radius r is 5.

• S3

  Substitute these values into the equation: (x - 3) 2 + (y + 4) 2 = 5 2 .

• S4

  Simplify the right-hand side: (x - 3) 2 + (y + 4) 2 = 25.

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   The standard form of the equation of a circle is (x - a) 2 + (y - b) 2 = r 2 .

2. S2

   Here, the centre (a, b) is (3, -4) and the radius r is 5.

3. S3

   Substitute these values into the equation: (x - 3) 2 + (y + 4) 2 = 5 2 .

4. S4

   Simplify the right-hand side: (x - 3) 2 + (y + 4) 2 = 25.

5. Final answer: (x - 3) 2 + (y + 4) 2 = 25

Common mistakes

• Forgetting to complete the square when finding the centre and radius. — Always rewrite the equation in the form (x - a) 2 + (y - b) 2 = r 2 by completing the square for both x and y terms.
• Incorrectly calculating the radius when completing the square. — After completing the square, ensure you take the square root of the constant term on the right-hand side to find the radius.
• Misapplying the property that the angle in a semicircle is a right angle. — Always verify that the triangle is inscribed in a semicircle and that the angle at the circumference is opposite the diameter.
• Forgetting that the perpendicular from the centre to a chord bisects the chord. — Remember that the perpendicular from the centre to a chord always bisects the chord. Use this property to find the length of the chord or its midpoint.
• Incorrectly identifying the relationship between the radius and the tangent. — Always remember that the radius at any point on the circumference is perpendicular to the tangent at that point. Use this property to find angles or distances.
• Failing to check if a given equation represents a circle. — Always verify that the equation can be written in the standard form (x - a) 2 + (y - b) 2 = r 2 . If it cannot, the given equation may represent another conic section.
• Incorrectly calculating the midpoint of a chord. — Double-check your calculations and ensure you correctly apply the midpoint formula: (x 1 + x 2 ) / 2, (y 1 + y 2 ) / 2.

Where the marks go

• Full worked solution (all marking points)3 marks

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