IP AMaths Notes (Upper Sec, Year 3-4): 07) Parabolas and Circles

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Standard forms and locus techniques for conic questions in IP Additional Mathematics.

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Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 07) Parabolas and Circles cover?
A: Standard forms and locus techniques for conic questions in IP Additional Mathematics.

Upper-sec AMaths emphasises the ability to flip between algebraic equations and geometric interpretations of parabolas and circles.

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These forms sit within the SEAB GCE O-Level Additional Mathematics (4049) coordinate-geometry scope for circles. The parabola focus-directrix form is an extension to support locus reasoning; the core syllabus treats quadratic graphs primarily in the vertical $y = ax^{2} + bx + c$ orientation.

Status: SEAB O-Level Additional Mathematics 4049 syllabus (exams from 2025) checked 2025-11-30 - scope unchanged; remains the reference for this note.

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

Given information	Best first move	What to check
Centre and radius are visible	Write $(x - a)^2 + (y - b)^2 = r^2$ .	Keep signs opposite inside the brackets: centre $(a, b)$ gives $x - a$ and $y - b$ .
Three points on a circle	Use $x^2 + y^2 + Dx + Ey + F = 0$
Centre lies on an axis or line	Let the unknown centre be $(a, 0)$ , $(0, b)$ , or a point on the given line, then equate distances.	Square distances only after confirming both distances are radii.
Tangent to a circle	Radius to tangent point is perpendicular to the tangent.	Find the radius gradient first, then use the perpendicular gradient for the tangent.
Tangent to a parabola	Find the point first, then differentiate or use the given tangent condition.	Substitute the tangent point back into the original parabola.

Step	What to do	Common trap
1	Group $x$ -terms and $y$ -terms separately.	Mixing the $x$ and $y$ completions into one bracket.
2	For $x^2 + Dx$ , add and subtract $\left(\dfrac{D}{2}\right)^2$ .	Halving $D$ but forgetting to square the half.
3	Move the leftover constant to the other side.	Reading the radius from a negative or unsimplified right-hand side.
4	Read centre from opposite signs inside brackets.	Saying $(x+4)^2$ gives centre $(4,0)$ instead of $(-4,0)$ .

IP AMaths Notes (Upper Sec, Year 3-4): 07) Parabolas and Circles

Sources

Choosing the conic route

1 Standard forms

Circle completing-square checkpoint

2 Worked example - Circle from three points

3 Worked example - Tangent to a parabola

4 Worked example - Circle with centre on the $x$ -axis

5 Practice Quiz

6 Try this

Related Posts

Sources

Choosing the conic route

1 Standard forms

Circle completing-square checkpoint

2 Worked example - Circle from three points

3 Worked example - Tangent to a parabola

4 Worked example - Circle with centre on the x x x-axis

5 Practice Quiz

6 Try this

Related Posts

4 Worked example - Circle with centre on the $x$ -axis